Proving that PP is closed under symmetric difference
As stated in the Wikipedia article about PP, a proof that PP is closed under symmetric difference was a PhD thesis by David Russo Structural properties of complexity classes. Proving that PP is closed under union was also open for a long time until it was answered by R. Beigel, N. Reingold, and D. A. Spielman, "PP is closed under intersection", Proceedings of ACM Symposium on Theory of Computing 1991