The logarithmic Brunn-Minkowski conjecture for origin-symmetric convex bodies

Let $K,L\in\mathcal{K}_e(\mathbb{R}^n)$ be origin-symmetric convex bodies containing the origin in their interiors, and let $K^tL^{1-t}$ denote their logarithmic combination, defined as the Wulff shape of $h_K^th_L^{1-t}$, where $h_K$ and $h_L$ are the support functions. For $t\in[0,1]$, The logarithmic Brunn-Minkowski conjecture.

$$\operatorname{vol}(K^tL^{1-t})\geq \operatorname{vol}(K)^t\operatorname{vol}(L)^{1-t}.$$

This is the logarithmic analogue of the Brunn-Minkowski inequality and is known in dimension two, for bodies sufficiently close to the unit $\ell^q$-ball when $q\geq 2$, and under certain reflection symmetries. The general case remains open.