The log-Brunn-Minkowski conjecture for centrally symmetric convex bodies

Let $K,L\subset\mathbb R^n$ be centrally symmetric convex bodies, let $\lambda\in[0,1]$, and define their log-Minkowski combination by

$$(1-\lambda)K+_o\lambda L=\bigcap_{u\in S^{n-1}}\{x:\langle x,u\rangle\le h_K(u)^\lambda h_L(u)^{1-\lambda}\}.$$

The log-Brunn-Minkowski conjecture. One has

$$\operatorname*{vol}((1-\lambda)K+_o\lambda L)\ge \operatorname*{vol}(K)^{1-\lambda}\operatorname*{vol}(L)^\lambda.$$

Böröczky, Lutwak, Yang and Zhang showed this conjecture in the plane; it also holds for unconditional bodies. It implies the $L^p$-Brunn-Minkowski conjecture for every $p>0$, but remains open in general.