The B-conjecture for even log-concave measures

Let $\mu$ be an even log-concave measure on $\mathbb{R}^n$, and let $K$ be a symmetric convex body.

B-conjecture. The function

$$\mathbb{R}\ni t\longmapsto \mu(e^tK)$$

is log-concave.

This conjecture is closely connected with the logarithmic Brunn–Minkowski conjecture. The supplied status evidence records that the Gaussian case was confirmed, while the full formulation is presented in the paper as a conjecture.