If there is a winning strategy, is it for White? We do not know, given two perfect players White and Black, whether the game would necessarily end in a draw, or necessarily end in a win (for either Black or White).

However, can we prove that if there is a winning strategy, then it is for White? In other words, can we prove that Black must either lose or draw?
If there is such a proof, no one has found it, and I doubt very much that such a proof exists (it is hard to imagine a mathematically provable "guaranteed-draw" strategy as White). One would certainly expect White to have an advantage if anybody did, but there are some downsides to going first as well (you have to disclose information before your opponent), so it is theoretically possible that the downsides outweigh the upsides. That said, the probability of that being the case seems to be infinitesimal.