The local $L^p$-Brunn-Minkowski inequality for $C^2_+$ bodies

Let $K$ be a $C^2_+$ centrally symmetric convex body, let $p\in[0,1)$, and let $z\in C^2(S^{n-1})$ be even. For a function $f$ on the sphere, write $f[k]$ for its occurrence $k$ times in a mixed-volume expression, and let $V$ denote mixed volume.

The local $L^p$-Brunn-Minkowski inequality. One has

$$\frac{1}{\operatorname*{vol}(K)}V(z h_K[1],K[n-1])^2\ge\frac{n-1}{n-p}V(z h_K[2],K[n-2])+\frac{1-p}{n-p}V(z^2h_K[1],K[n-1]).$$

Kolesnikov and Milman showed implications between this local inequality and the global $L^p$-Brunn-Minkowski inequality, and proved the local result for $p$ sufficiently close to $1$. The source records their conjecture that the local and global inequalities are equivalent.