Higher-dimensional Cheeger log-convexity conjecture

Let $K\subset\mathbb{R}^n$ be a convex body, let $P>0$, and let $h(K)=\inf_{E\subset K}\operatorname{Per}(E)/|E|$ be its Cheeger constant. Higher-dimensional Cheeger log-convexity conjecture. The function

$$t\longmapsto h(P^tK)$$

is log-convex in arbitrary dimension. Log-convexity is proved in dimension $2$ in the source; the conjecture asks for the corresponding statement in every dimension.