Symmetric convex-subset conjecture for log-concave isoperimetric sets

Let $0<\alpha<1$, let $\mu=w(x)dx$ be a strictly log-concave probability measure supported on all of $\mathbb{R}^n$, with centrally symmetric density $w(-x)=w(x)$, and let $E$ be an isoperimetric subset with $\mu(E)=\alpha$. Symmetric convex-subset conjecture. There is $b=b(\alpha)>0$ and a convex set $\Gamma$ such that

$$\mu(\Gamma)\ge b(\alpha),\qquad \Gamma\subset E,\qquad -\Gamma\subset E^c=\mathbb{R}^n\setminus E.$$

This is a symmetry-breaking variant of the convex-sandwich conjecture for general volume fractions and centrally symmetric densities.