This is similar to the alternate reality counter proof: If alternate realities exist, and we can get to them from this reality, then they are no longer an alternate reality. If we cannot get to them, then they cannot "exist" as our reality is a prerequisite for existence.
>If alternate realities exist, and we can get to them from this reality, then they are no longer an alternate reality.
All that means is that they are somehow connected.
They are still 2 different environments, which different rules -- which is what people actually mean when they say "alternate reality", and not "complete isolated reality space".
So the counter-proof is more like saying "If there's Canada, and we get to there from here, the they are no longer Canada".
It the hypothetical example described it could be anything -- in fact it could be exactly like the "borders of Canada", some stretch of land with guards and fences. Or it could be some "portal", a mirror you can go through, a black hole, whatever.
That place would just have to comply with a different set of rules compared to anywhere else to be justified to call it an "alternate reality" (e.g. no gravity, fire is cold, etc).
Heck, we even call experiences like living with the Amish an "alternative reality", and those are on our very same universe/world, and with all the known laws of physics and basics intact.
That argument is getting hung up on semantics. It's like saying north and south America are the same continent because they are connected. It is implicitly choosing a very specific definition of reality amongst many possible definitions.
The thing is that "personality" is a genuinely contested concept in the field (assuming that you go beyond the DSM "axes" and look at the broader debate). The semantic mess isn't merely obfuscation of established knowledge, it's also a reflection of the state of the art. From what I understand, the Big Five model [1] has the strongest empirical validation, but there are also many experts in personality psychology who regard that model as an oversimplification or even perversion of more sophisticated ideas (cf. the conflict between adherents of insight-oriented/psychodynamic versus cognitive-behavioral modes of therapy). See also [2] to get a sense of how varied the concepts are.
So screw that concept and talk about ones that are useful - ones that can allow us to make predictions. Can a person who is now bigoted and prone to violence become a Gandhi? I'd say yes, I've seen similar thing happen. Call it "personality change" or call it something else, but the phenomenon seems to be real.
You can say the same for a changed personality :p