And what do you imagine inflationary theory "predicts"? What is its "empirical" foundation? It "predicts" nothing in the manner that one typically would understand this term outside of certain areas within physics, because the models are constructed based upon known physics and observations and then unknown physics (e.g., and in particular, "inflatons") are introduced in a manner that allows us to produce a wide-variety of models that can yield any predictions we like. Its "success" is primarily in the ability to produce such models.
So, in short, inflationary models are constructed by taking what we know from observations and known physics and the resulting cosmology, and then introducing physics that we not only have no evidence for, but that we also don't actually understand beyond the features that we select (e.g., scalar-valued quantum fields that we demand to cause certain results to produce inflation in the manner we would like).
We can then look at the variety of models that can "predict" what we already know, and see what other features seem to be demanded alongside. It is at this stage, for the most part, that certain multiversal claims are supposedly substantiated by the sorts of models that yield the observed imbalances, inhomogeneities, flatness, etc., and that do so in a manner that many prefer to e.g., the anthropic principle or simply just accepting that what we see is what we get rather than demanding an explanation for what appears to be an otherwise unnatural and/or finely-tuned set of small parameters necessary to explain the universe find ourselves in.
So, these are "well-supported" only if one accepts a fundamental premise for which inflationary cosmology was introduced in the first place: that the universe appears to be highly improbable, unnatural, and finely-tuned in a manner that should be explained by some sort of physical "mechanism" that we lack any evidence or support for. That inflationary models tend to suggest multiverse cosmologies ought to be understood as a weakness of our basic lack of any understanding of the physics that we are proposing to have taken place in order to cause the inflation that is itself supported by the desire to produce the observed universe in a manner that does not rely on fine-tuning or shrugging off the nearly impossible probabilities that the observed universe is as we find it by chance.
This, again, is quite similar to the kind of "argument from design" used by creationists and the like. Sure, it has more mathematics and unlike arguments from design we can examine the different scenarios that result from altering the distributions of various parameters that one cannot by asserting "God did it", but this sort of speculation remains exactly this until either one accepts that 1) a seemingly unnaturally finely-tuned universe requires some kind of explanation or 2) we find some sort of actual empirical support for inflation, such as a plausible physical theory that can be tested rather than introducing yet another dreamt up quantum field we've never encountered and don't understand but, like the "drainons" of the string theoretic swampland, are supported by yielding results in models that we like (in the case of "drainons", it is the draining of the swampland approach to string landscapes; in the case of "inflatons", it is inflation)..