The way you have presented it, it is contrary to LDS teachings. A great deal can happen in the Spirit World. In the end, it's not going to be a matter of LDS versus non-LDS. There are millions upon millions of non-LDS people living in the world today who are going to end up in the Celestial Kingdom and proportionately just as many LDS people who will not end up in the Celestial Kingdom. It doesn't work the way you're describing it.
Keep His commandments.
This is what I've gleaned from my studies(again, correct me if I'm wrong):
In the Spirit World, the gospel (LDS version which LDS claim is not their version, but the Lord’s) is presented by
worthy LDS to those who were not righteous enough to make it to Paradise but were instead sent to Spirit Prison. The
unrighteous LDS will not need to hear it because they already heard it in this life. Spirit Prison is a temporary hell which gives people the opportunity to either suffer and repent or suffer and not repent.
Those people who accept the LDS gospel will need proxy temple work done on their behalf on earth because the ordinances cannot be done in Spirit Prison. However, Doctrine and Covenants 132, verse 18 says:
"...
when they are out of the world it cannot be received there, because the angels and the gods are appointed there, by whom they cannot pass; they cannot, therefore, inherit my glory; for my house is a house of order, saith the Lord God."
Doctrine and Covenants 132
15 Therefore, if a man marry him a wife in the world, and he marry her not by me nor by my word, and he covenant with her so long as he is in the world and she with him, their covenant and marriage are not of force when they are dead, and when they are out of the world; therefore, they are not bound by any law when they are out of the world.
16 Therefore, when they are out of the world they neither marry nor are given in marriage; but are appointed angels in heaven, which angels are ministering servants, to minister for those who are worthy of a far more, and an exceeding, and an eternal weight of glory.
17 For these angels did not abide my law; therefore, they cannot be enlarged, but remain separately and singly, without exaltation, in their saved condition, to all eternity; and from henceforth are not gods, but are angels of God forever and ever.
18 And again, verily I say unto you, if a man marry a wife, and make a covenant with her for time and for all eternity, if that covenant is not by me or by my word, which is my law, and is not sealed by the Holy Spirit of promise, through him whom I have anointed and appointed unto this power, then it is not valid neither of force when they are out of the world, because they are not joined by me, saith the Lord, neither by my word; when they are out of the world it cannot be received there, because the angels and the gods are appointed there, by whom they cannot pass; they cannot, therefore, inherit my glory; for my house is a house of order, saith the Lord God.
19 And again, verily I say unto you, if a man marry a wife by my word, which is my law, and by the new and everlasting covenant, and it is sealed unto them by the Holy Spirit of promise, by him who is anointed, unto whom I have appointed this power and the keys of this priesthood; and it shall be said unto them—Ye shall come forth in the first resurrection; and if it be after the first resurrection, in the next resurrection; and shall inherit thrones, kingdoms, principalities, and powers, dominions, all heights and depths—then shall it be written in the Lamb’s Book of Life, that he shall commit no murder whereby to shed innocent blood, and if ye abide in my covenant, and commit no murder whereby to shed innocent blood, it shall be done unto them in all things whatsoever my servant hath put upon them, in time, and through all eternity; and shall be of full force when they are out of the world; and they shall pass by the angels, and the gods, which are set there, to their exaltation and glory in all things, as hath been sealed upon their heads, which glory shall be a fulness and a continuation of the seeds forever and ever.
20 Then shall they be gods, because they have no end; therefore shall they be from everlasting to everlasting, because they continue; then shall they be above all, because all things are subject unto them. Then shall they be gods, because they have all power, and the angels are subject unto them.
21 Verily, verily, I say unto you, except ye abide my law ye cannot attain to this glory.
22 For strait is the gate, and narrow the way that leadeth unto the exaltation and continuation of the lives, and few there be that find it, because ye receive me not in the world neither do ye know me.
23 But if ye receive me in the world, then shall ye know me, and shall receive your exaltation; that where I am ye shall be also.
24 This is eternal lives—to know the only wise and true God, and Jesus Christ, whom he hath sent. I am he. Receive ye, therefore, my law.
25 Broad is the gate, and wide the way that leadeth to the deaths; and many there are that go in thereat, because they receive me not, neither do they abide in my law.
26 Verily, verily, I say unto you, if a man marry a wife according to my word, and they are sealed by the Holy Spirit of promise, according to mine appointment, and he or she shall commit any sin or transgression of the new and everlasting covenant whatever, and all manner of blasphemies, and if they commit no murder wherein they shed innocent blood, yet they shall come forth in the first resurrection, and enter into their exaltation; but they shall be destroyed in the flesh, and shall be delivered unto the buffetings of Satan unto the day of redemption, saith the Lord God.
27 The blasphemy against the Holy Ghost, which shall not be forgiven in the world nor out of the world, is in that ye commit murder wherein ye shed innocent blood, and assent unto my death, after ye have received my new and everlasting covenant, saith the Lord God; and he that abideth not this law can in nowise enter into my glory, but shall be damned, saith the Lord.
James E. Talmage wrote:
In the merciful providence of the Almighty, provision has been made for vicarious service by the living for the dead, in the ordinances essential to salvation; so that all who in the spirit-world accept the word of God as preached to them, develop true faith in Jesus Christ as the one and only Savior, and contritely repent of their transgressions, shall be brought under the saving effect of baptism by water for the remission of sins, and be recipients of the baptism of the Spirit or the bestowal of the Holy Ghost.
James E. Talmage, Jesus the Christ, p. 675
The unrighteous LDS are pretty much up a creek:
"The assumption that the gracious assurance given by Christ to the penitent sinner on the cross was a remission of the man’s sins, and a passport into heaven, is wholly contrary to both the letter and spirit of scripture, reason, and justice. Confidence in the efficacy of death-bed professions and confessions on the basis of this incident is of the most insecure foundation. The crucified malefactor manifested both faith and repentance; his promised blessing was that he should that day hear the gospel preached in paradise; in the acceptance or rejection of the word of life he would be an agent unto himself. The requirement of obedience to the laws and ordinances of the gospel as an essential to salvation was not waived, suspended, or superseded in his case."
James E. Talmage, Jesus the Christ, Chapter 36, p. 677
Does the following apply to only LDS or to all men?
Book of Mormon, Alma 34
32 For behold, this life is the time for men to prepare to meet God; yea, behold the day of this life is the day for men to perform their labors.
33 And now, as I said unto you before, as ye have had so many witnesses, therefore, I beseech of you that ye do not procrastinate the day of your repentance until the end; for after this day of life, which is given us to prepare for eternity, behold, if we do not improve our time while in this life, then cometh the night of darkness wherein there can be no labor performed.
34 Ye cannot say, when ye are brought to that awful crisis, that I will repent, that I will return to my God. Nay, ye cannot say this; for that same spirit which doth possess your bodies at the time that ye go out of this life, that same spirit will have power to possess your body in that eternal world.
35
For behold, if ye have procrastinated the day of your repentance even until death, behold, ye have become subjected to the spirit of the devil, and he doth seal you his; therefore, the Spirit of the Lord hath withdrawn from you, and hath no place in you, and the devil hath all power over you; and this is the final state of the wicked.