Even log-concave Brunn–Minkowski conjecture for origin-symmetric convex sets

Let $\nu$ be an even log-concave measure on $\mathbb{R}^{n}$. Let $K_{0},K_{1}\subseteq\mathbb{R}^{n}$ be convex and origin-symmetric, meaning $K_{0}=-K_{0}$ and $K_{1}=-K_{1}$. Even log-concave Brunn–Minkowski conjecture. For every $t\in[0,1]$,

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

This generalizes the sharp Gaussian inequality for origin-symmetric convex sets to all even log-concave measures. The source describes it as widely believed and gives no resolution, so it remains open.