If Bob knows that the particle becomes entangled with him, then he still makes the same predictions.

Ok, that's surprising. Here's why I thought otherwise: From Bob's perspective, a particle is prepared in a superposition of states B and C. Then Bob observes or becomes entangled with the particle, thus collapsing its state. Then the super-duper quantum erasure is performed, which preserves the state of the particle. Then the particle strikes the second half-silvered mirror. A collapse interpretation tells Bob to expect two outcomes with equal probability. Is this, then, an experiment where a collapse interpretation and a many-worlds interpretation give different predictions?