The symmetric $L^p$ Brunn–Minkowski conjecture

Let $K$ and $L$ be symmetric convex bodies in $\mathbb{R}^n$, let $V$ denote volume, and let $\lambda\cdot K+_p(1-\lambda)\cdot L$ denote their $L^p$-Minkowski combination. The symmetric $L^p$ Brunn–Minkowski conjecture. If $0<p<1$ and $0<\lambda<1$, then

$$V(\lambda\cdot K+_p(1-\lambda)\cdot L)^{\frac{p}{n}}\geq \lambda V(K)^{\frac{p}{n}}+(1-\lambda)V(L)^{\frac{p}{n}}.$$

This extends the classical and known $L^p$ Brunn–Minkowski inequalities to the range $0<p<1$ for symmetric convex bodies; the supplied material gives no resolution.