The Symmetric Gaussian Problem

Let $n$ be a positive integer, let $0<a,b<1$, and let $-1\leq\rho\leq1$. For measurable sets $A,B\subseteq\mathbb{R}^{n}$, write $\gamma_n(A)=a$ and $\gamma_n(B)=b$, and let $r_a,r_b,r_a',r_b'>0$ satisfy

$$\gamma_n(B(0,r_a))=a,\quad \gamma_n(B(0,r_b)^c)=b,\quad \gamma_n(B(0,r_a')^c)=a,\quad \gamma_n(B(0,r_b'))=b.$$

Symmetric Gaussian Problem. If $\rho>0$, then either $(B(0,r_a),B(0,r_b)^c)$ or $(B(0,r_a')^c,B(0,r_b'))$ achieves

$$\inf_{\substack{A,B\subseteq\mathbb{R}^{n}:\\ \gamma_n(A)=a,\,\gamma_n(B)=b,\,A=-A}}\int_{\mathbb{R}^{n}}1_A(x)T_\rho(1_B)(x)\,d\gamma_n(x).$$

If $\rho<0$, the same assertion holds with the additional restriction $B=-B$. This problem asks for the extremal symmetric sets for Gaussian noise correlation and is motivated by connections to communication complexity and related concentration inequalities; the source presents it as a conjecture and gives no resolution.