Lipschitz-interface conjecture for isoperimetric sets of strictly log-concave measures

Let $\mu=e^{-V}dx$ be a strictly log-concave probability measure on $\mathbb{R}^n$, and let $E$ be an open isoperimetric subset with $\mu(E)=\alpha$. Lipschitz-interface conjecture. For suitable growth conditions on $V$, including bounded convex domains in the limiting case, there is a constant $C=C(\alpha,n)$ and a rotation after which

$$\partial E=\{(x',x_n)\in\mathbb{R}^{n-1}\times\mathbb{R}:x_n=\varphi(x')\},\qquad |\nabla\varphi(x')|\le C(\alpha,n).$$

The claim extends the proposed convex-domain interface regularity to strictly log-concave densities; the source leaves the precise admissible growth conditions on $V$ to be specified.