Convex sandwich conjecture for isoperimetric sets of strictly log-concave measures

Let $\mu$ be a finite, strictly log-concave measure on $\mathbb{R}^n$, and let $E$ be an isoperimetric set with volume fraction $0<\alpha<1/2$. Convex sandwich conjecture. There are convex sets $G_1$ and $G_2$ such that $G_1\subset E\subset G_2$ and

$$\mu(G_1)\ge c(\alpha,n)\mu(\mathbb{R}^n),\qquad \mu(G_2)\le(1-c(\alpha,n))\mu(\mathbb{R}^n),$$

where $c(\alpha,n)>0$ can be chosen independently of $n$. This is the strictly log-concave analogue of the convex-hull formulation and aims to establish quantitative convexity on both sides of the interface.