Measure-valued extension of the log-Brunn–Minkowski inequality

Let $p \in [0,1]$. Let $\mu$ be a symmetric measure in $\mathbb R^n$ with an $\alpha$-concave density function, where $\alpha \geq -\frac{p}{n}$. Let $\oplus_p$ denote the $p$-Firey combination used in the source, and let $M_s^\lambda$ denote the $s$-mean. Measure-valued log-Brunn–Minkowski extension. For every pair of symmetric convex bodies $K,L$ in $\mathbb R^n$ and every $\lambda \in [0,1]$,

$$\mu((1-\lambda) \cdot K \oplus_p \lambda \cdot L) \geq M_{\left(\frac{n}{p}+\frac{1}{\alpha}\right)^{-1}}^\lambda(\mu(K),\mu(L)).$$

This is presented as an extension proposed by the author in earlier work. The supplied passage does not establish the inequality or provide a resolution status, and the notation for the $p$-combination and generalized mean is only referred to rather than fully defined here.