Borisenko's isoperimetric conjecture for ball bodies

Let $n\in\mathbb N$ and let $V\in(0,\kappa_n)$. For $K\in\mathcal S_n$, fix the volume ${\rm Vol}(K)=V$.

Borisenko's conjecture. Among all such sets, the sets maximizing surface area are precisely the lenses of volume $V$.

This conjecture concerns the extremal surface area of ball bodies at fixed volume. It was suggested by Borisenko, first appearing in Drach's doctoral dissertation and later formally stated in English; its resolution is not specified here.