Variance-optimal functions determine isoperimetric extremizers

Let $G$ be a graph, let $X$ be a variance-optimal function over $G$, and for suitable $r$ define $S_r=\{a=(a_1,\ldots,a_n)\in V(G^n):\sum_{i=1}^n X(a_i)\leq r\}$. For a vertex set $S\subseteq V(G^n)$, let $B_d(S)=\{u:d_{G^n}(u,S)\leq d\}$.

Isoperimetric extremizer conjecture. For $n$ sufficiently large and $d,r$ in appropriate ranges,

$$|B_d(S_r)|\leq |B_d(S')|$$

for every $S'\subset V(G^n)$ with $|S_r|\leq |S'|$.

This conjecture proposes that sets formed from variance-optimal functions determine the extremal isoperimetric behavior of Cartesian powers of graphs. The supplied text gives no resolution or precise ranges for $d$ and $r$.