Two hyperplane conjecture for centrally symmetric convex bodies

Let $\Omega\subset\mathbb{R}^n$ be convex, bounded, and centrally symmetric, and let $E$ be an isoperimetric subset of $\Omega$ with $|E|/|\Omega|=1/2$. Two hyperplane conjecture. There is an absolute constant $c>0$ such that (a) there is a convex cone $\Gamma$ with

$$\Omega\cap\Gamma\subset E,\qquad \Omega\cap(-\Gamma)\subset\Omega\setminus E,\qquad |\Gamma\cap\Omega|\ge c|\Omega|.$$

(b) The cone $\Gamma$ can be taken to be a half-space. This is presented as a strong, dimension-independent trapping statement for the interface and is related to the Kannan–Lovász–Simonovits hyperplane conjecture.