Borisenko's reverse isoperimetric conjecture for lambda-convex bodies

Let $n\geqslant 2$, and let $M^n(c)$ be an $n$-dimensional model space of constant sectional curvature $c\in\mathbb{R}$. Let $K\subset M^n(c)$ be a $\lambda$-convex body, and let $L\subset M^n(c)$ be a $\lambda$-convex lens. For a compact submanifold with boundary $A^m\subset M^n(c)$, write $|A|$ for its $m$-dimensional volume. Borisenko's conjecture. If $|\partial K|=|\partial L|$, then $|K|\geqslant |L|$, with equality if and only if $K$ is a $\lambda$-convex lens. This is a reverse isoperimetric problem for $\lambda$-convex bodies. The paper states that the conjecture is confirmed in three-dimensional model spaces of constant curvature for $c\neq 0$, while it complements recent progress in the Euclidean case $c=0$; as presented here, the general conjecture remains open.