I'm not very familiar with multi-core CPUs, but I think the question may be meaningless. I don't think you'll be switching CPUs without switching execution threads. Essentially, each core does have its own ax register, but it is saved and restored when tasks are switched, so you'll always have "your own" ax register to work with. I don't think it's anything you need to worry about, unless you're writing your own multi-tasking OS.
I think that the "true" answer is more complicated than that, but I hope that'll suffice. Corrections welcomed!
Best,
Frank