Gardner–Zvavitch conjecture for even log-concave measures

Let $\nu$ be an even log-concave measure on $\mathbb{R}^{n}$. For origin-symmetric convex bodies $K_{0},K_{1}\subseteq\mathbb{R}^{n}$ and $t\in[0,1]$, define

$$(1-t)K_{0}+tK_{1}=\{(1-t)x_{0}+tx_{1}:x_{0}\in K_{0},\ x_{1}\in K_{1}\}.$$

The measure $\nu$ is log-concave when it has a density of the form $d\nu/dx=e^{-V}$ with $V$ convex. Gardner–Zvavitch conjecture. For every such $\nu$, $K_{0}$, $K_{1}$, and $t$, one has

$$\nu\left((1-t)K_{0}+tK_{1}\right)^{\frac{1}{n}}\geq(1-t)\nu(K_{0})^{\frac{1}{n}}+t\nu(K_{1})^{\frac{1}{n}}.$$

This conjecture extends the dimensional Brunn–Minkowski inequality from Gaussian measure to all even log-concave measures. The Gaussian case was conjectured by Gardner and Zvavitch and was proved by Eskenazis and Moschidis; the general even log-concave case remains open.