The quadratic Symmetric Gaussian Problem

Let $n$ be a positive integer, let $0<a<1$, and let $-1\leq\rho\leq1$. For a measurable set $A\subseteq\mathbb{R}^{n}$ with $\gamma_n(A)=a$, choose $r_a,r_a'>0$ such that

$$\gamma_n(B(0,r_a))=a,\qquad \gamma_n(B(0,r_a')^c)=a.$$

Quadratic Symmetric Gaussian Problem. Either $B(0,r_a)$ or $B(0,r_a')^c$ achieves

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

This is the special case obtained by restricting the Symmetric Gaussian Problem to $a+b=1$ and $A=B^c$; the source provides no evidence that this quadratic version has been resolved.