Could CPT symmetry explain why there's no antimatter?

I've been wondering if the Big Bang wasn't the "beginning" of time like we usually think, but a single event exhibiting charge-parity-time (CPT) symmetry, which happened at a particular moment in time when the universe had minimum entropy. In that scenario some sort of singularity with a neutral charge would explode into "two" universes: this universe in which we find ourselves, time flowing along a positive time axis, and a twin universe with all the antimatter in it, flowing along a negative time axis.

Each universe would have its own distinct thermodynamic arrow of time, pointing in opposite directions, so that antimatter aliens in the "past" universe would, just like us, be viewing the Big Bang as something that happened in their past- since the Big Bang was a point in time with minimum entropy. (They would agree that heat energy flows from hot things to cold things if you wait. But they are "waiting" into what we would consider their past.)

They would consider their universe to be expanding, just like we do, meaning the cosmological arrow of time would point in the same direction as the thermodynamic arrow of time in both universes. (So antimatter aliens wouldn't be terrified of a "Big Crunch", even if the Big Bang happened in what we here would think would be their "future".)

That would get us charge symmetry and time symmetry. I'm not sure of what sort of parity symmetry there might be, as in one universe going "left" and one going "right". (Maybe antimatter aliens are "really" left-handed, as we would consider it, but they would claim to be right-handed since they don't realize their eyes and brains are wired backwards too? I don't know.) But if there was some sort of parity symmetry involved with the Big Bang, it would complete CPT symmetry and explain why we don't see any antimatter.

Does that make any sense to anybody?