The $(p,q)$-conjecture for log-concave measures

Let $p\in[0,1]$, let $\mu$ be an even log-concave measure on $\mathbb R^n$, let $K,L$ be nonempty symmetric convex sets, and let $\lambda\in[0,1]$. For $p>0$, define

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

with $p=0$ interpreted by the zero-sum. The $(p,q)$-conjecture. For every $q\in[0,p]$, one should have

$$\mu(\lambda K+_p(1-\lambda)L)^{q/n}\geq\lambda\mu(K)^{q/n}+(1-\lambda)\mu(L)^{q/n}.$$

This unifies the $L_p$-Brunn–Minkowski and dimensional Brunn–Minkowski conjectures and is stated in the paper as following from the Log-Brunn–Minkowski conjecture. Its general validity remains open.