It's unclear what kind of answer you're expecting. You did a lengthy summary of the state of the art, from which it is obvious that it is not known for which $n$ optimal play results in a draw, and MO is not a place to formulate conjectures. The question “for which values of $n$ is it mathematically proven that optimal play results in a draw, resp. a win for some player?” is a valid one, but if so, you had better track your “citation needed” about $n=8$: where did you get this information from?
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