1. Particles are indistinguishable from each other. It's a very deep principle, i.e. a lot of stuff relies on this being true.
2. States and particles can absolutely be entangled ("bound") to each other, but it tends to be pretty fragile.
So it gets tossed on the stack with all the other complex-and-unfalsifiable theories for which no evidence exists.
It might make for an amusing sci-fi plot though.
The obviousness or lack thereof is subjective, but the exclusivity is firmly established. The absolute indistinguishability of particles is deeply woven into quantum mechanics; you don't get a Pauli exclusion principle without it, for example. If the particles remembered their previous lives, and an electron that used to be tied to an iron nucleus weren't completely identical to one that used to be stuck to a carbon nucleus, all of quantum mechanics as we know it would be impossible.
Experimentally you'd be attempting to detect inexplicable single particle events above some level of rarity. You'd have access to only one side of the pair - you can't tell which one the other side is even if it's right in front of you (and it almost certainly isn't). So there's no discernible (to you) trigger for these events you're trying to detect. So you'd be trying to correlate frequency counts with bulk conditions as averaged across more or less the entire universe.
In the same vein as the God of the gaps the phenomenon could always be hiding below the noise floor.