If the 'avatar' is convincing and can have a continued existence as a person independent of my interaction with it, i.e. I don't "have" them as a form of possession, yes. It does seem better than 50/50 to me though that some (maybe all) wouldn't want continued existence and would decide to go back to not existing (for all I can tell), there may even be strong predictive signs of that in the brain models such that they don't even need to be temporarily brought back and asked or first made to listen to arguments or just have some final-final talks with me/others before deciding. I'd accept that.

For less convincing avatars where the point is just my own benefit of conversing with something like them when I want, from slightly like them to eerily like them, for one it's a weak yes, for the others I'm more indifferent -- it'd be more in the realm of curiosity than desire, like talking to a historical figure or a fictional character. The weak yes I expect will get weaker (as it already has, despite non-linear flare-ups/resurgences where it's temporarily stronger) and eventually match the others after long enough.