Is every polyhedron with maximum volume for its surface area and number of faces convex? If so, does this apply even if all faces lying in the same plane are counted as one face? One can also ask this question in dimensions other than three.
A good resource about sphericity is Alan Schoen's geometry page Roundest Polyhedra. Citing older works written in German, it notes that maximum sphericity convex $n$-polyhedra (i.e. with $n$ faces) have an inscribed sphere tangent to all faces at their centroids. A proof of this would likely yield convexity as well.
Maximizing sphericity and related polyhedral properties given a fixed number of faces $n$ leads to a rich theory, both for small $n$, and especially asymptotically if we look closer at it than simply being sphere-like, as I describe in my draft paper Asymptotic Optimal Sphericity.