Poincaré–Korn rigidity conjecture for strongly log-concave measures

Let $\mu$ be a centered and isotropic probability measure on $\mathbb{R}^n$ with density $d\mu=e^{-\phi}dx$, meaning

$$\int x\,d\mu=0,\qquad \int xx^T\,d\mu=\operatorname{Id}.$$

Let $\gamma$ denote the standard Gaussian measure, and let $C_{PK}$ be the Poincaré–Korn constant. Poincaré–Korn rigidity conjecture. If

$$\nabla^2\phi\geq\operatorname{Id},$$

then $C_{PK}(\mu)\geq C_{PK}(\gamma)$, with equality only if $\mu=\gamma$. This conjecture concerns rigidity and optimality of the standard Gaussian among centered isotropic measures satisfying a uniform convexity condition; the supplied context does not state whether it has been resolved.