The following problem is listed in Steven Lay's "Convex Sets and Their Applications" (1982) as unsolved (paraphrased):

Let $B$ be the unit ball in $\mathbb{R}^3$ and $0 < V < \pi$. Define $\mathcal{F}$ as the family of all convex subsets of $B$ with volume $V$. Find the member of $\mathcal{F}$ with maximum surface area.

The conjectured answer (as of 1982) is $B \cap$ { $(x_1,x_2,x_3)\in \mathbb{R}^3 | |x_1| < c $} for appropriately chosen $c$.

Does anyone know if this problem has been solved, or if any progress has been made on it? I couldn't find any recent references in the literature to it.