The unconditional log-Brunn–Minkowski conjecture for symmetric convex bodies

Let $K$ and $L$ be symmetric convex bodies in $[?]$; write $V(K)$ for volume and, for $0<p<1$, let $[?]$ denote the $L^p$-Minkowski combination. For $p=0$, define the 0-convex combination by its support function, so that

$$\lambda\cdot K+_o(1-\lambda)\cdot L=\{x\in\mathbb{R}^n\mid x\cdot u\leq h_K^\lambda(u)h_L^{1-\lambda}(u)\text{ for all }u\in S^{n-1}\}.$$

The unconditional log-Brunn–Minkowski conjecture. Inequality $V(\lambda\cdot K+_o(1-\lambda)\cdot L)\geq V(K)^\lambda V(L)^{1-\lambda}$ is true for all $\lambda\in[0,1]$ and all symmetric convex bodies $K,L$ in $\mathbb{R}^n$. This is the limiting $p=0$ form of the $L^p$ Brunn–Minkowski inequality; the question concerns whether this multiplicative volume inequality holds universally in the symmetric setting.