The local $L^p$-Brunn-Minkowski conjecture

Let $K,L\in\mathcal{K}_e(\mathbb{R}^n)$ be origin-symmetric convex bodies containing the origin in their interiors. Let $V$ denote the mixed volume functional, let $S_K$ be the surface area measure of $K$, let $h_K$ and $h_L$ be the support functions, and let $K[n-1]$ and $K[n-2]$ indicate repeated arguments in the mixed volume. For all $p\geq 0$, The local $L^p$-Brunn-Minkowski conjecture.

$$\frac{V(L,K[n-1])^2}{\operatorname{vol}(K)}\geq \frac{n-1}{n-p}V(L,L,K[n-2])+\frac{1-p}{n(n-p)}\int_{S^{n-1}}\frac{h_L^2}{h_K}dS_K.$$

This local inequality is equivalent to the $L^p$-Brunn-Minkowski inequality for $p\in[0,1)$. The corresponding conjecture was confirmed in that range, while the displayed assertion for all $p\geq0$ is not established in general.