First, let me clear something up. My 'proofs' of the No Infinities Principle aren't proofs. They're more like suggestions; the NIP is an empirical principle. We find regularities in nature. In particular, I find that every time a physicist's equation shows up an infinity, it doesn't mean there's an singularity in nature; it just means one of their assumptions, usually a simplifying assumption, was wrong. There is no black hole singularity and there is no Big Bang singularity.
As I said originally, physicists already use this principle, and they're doing so again, they're just not consistent about it. In this piece, Loop Quantum Cosmology replaces the singularity of the Big Bang with a Planck density, exactly as the NIP predicts. This Planck density will also apply to black holes, though since the ball of matter at this density can form an event horizon larger than its radius, we should still expect regular black holes.
(Note that this is also a serious problem for any possible beginning of our universe. It would be behind its own event horizon. This is partly why you can a priori validate inflation; something must be there to stop the Big Bang from immediately forming a universe-sized black hole. No, you don't get out of this by presuming this is what the inside of a black hole looks like; it simply repeats the problem. We would be starting with a point of infinite density inside a point of infinite density, forming a black hole inside a black hole, and ad nauseum.
(I think this is also the beginning of my proof that infinite regression is a fallacy. If it isn't a fallacy, you have to be able to resolve infinite series like these.
(Anyway, I suspect I know where the antimatter went. There is a centre of the universe, and though it's still not a preferred reference frame, there's a half-universe-sized antimatter black hole there; the inflation either doesn't work properly on antimatter, or it wasn't strong enough to pull all matter out of the Big Bang's initial configuration, and it just so happened that what we now call antimatter mostly remained behind. [Do note that the article, while it has good data, also has multiple errors in reasoning.])
The problem with LQC is that they don't realize eternal time is physically impossible.
Set t=(0) to be now. As this drifts into the past, [tick, t=(1), tick, t=(2)] set this moment t=(m). But time is relative, so let's make a coordinate transformation.
Lim m->(-∞) (m). What time is it now, specifically?
This operation is legal. This is what eternal, non-beginning time means. (Non-ending doesn't make any sense; the end is the present moment, though you'll notice the present is moving forward, and will continue to do so infinitely.) It means that time is meaningful no matter how far you go into the past, and since time is relative, you can set your temporal origin to any of those points.
What I immediately find is that for eternal time to be meaningful it cannot be relative. There must be a preferred origin. But we have empirically shown that time cannot be absolute, and we have a contradiction. So I can categorically predict that the 'bounce' will be found to erase all information. Our universe will be indistinguishable from one which began at that moment.
But let me be explicit. Once I've sent the origin infinitely far into the past, there is no transformation that can get it back to the present. Lim m->∞ (m) sends it infinitely far into the future, not to the present. We go from, relative to the origin, [t=(∞)] to [t=(-∞)] and time remains meaningless. (However, with an absolute origin, the present time will never actually be infinite.)
Hopefully when, as they discuss, they stop using general relativity as the jumping-off point for LQC, they will find something that prevents time from being infinite, because otherwise they're simply replacing one singularity with another of a different kind.