But in the absence of any human driven task that meaningfully improves well being the way to signal status will likely devolve into some form of fighting. The display of status might be narrowed to demonstrating who can kill more. Either literally kill or at best partake in some arbitrarily conceived games or intrigues designed (by AI?) to flag status.

Humans still successfully compete in chess status because other humans like watching humans compete.

I suspect it will be similar.

Hunger games vs. MeowMeowBeenz