The $L^p$-Brunn-Minkowski conjecture for centrally symmetric convex bodies

Let $K,L\subset\mathbb R^n$ be centrally symmetric convex bodies, let $p\in(0,1)$, and let $\lambda\in[0,1]$. Define the $L^p$-combination by its support-function description

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

The $L^p$-Brunn-Minkowski conjecture. One has

$$\operatorname*{vol}((1-\lambda)K+_p\lambda L)\ge (1-\lambda)\operatorname*{vol}(K)^{p/n}+\lambda\operatorname*{vol}(L)^{p/n}.$$

This is the conjectured subunit-exponent analogue of the Brunn-Minkowski inequality. The source notes that the conjecture is known in dimension two through the log-Brunn-Minkowski conjecture and that later work proves it for $p$ sufficiently close to $1$, while the general case remains open.