Separation-distance conjecture for least positive harmonic measures

For $d\geq3$, integers $n\geq2$ and $m\geq3$, define

$$\mathfs{M}_n^m(\mathbb{Z}^d):=\inf_{A\subset\mathbb{Z}^d,\,y\in A:\,\forall x_1,x_2\in A,\ \|x_1-x_2\|\geq m}\mathbb{H}_A(y),$$

where $\|x_1-x_2\|$ is graph distance. Separation-distance conjecture. (1) There exist $C(d,m),c(d,m)>0$ such that, for all $n\geq2$,

$$e^{-C n^{1/d}}\leq\mathfs{M}_n^m(\mathbb{Z}^d)\leq e^{-c n^{1/d}}.$$

(2) When $m\geq f(n)$ for some increasing function $f(n)$, there exist constants $C(d),c(d)>0$ such that, for all $n\geq2$,

$$cn^{-1}\leq\mathfs{M}_n^m(\mathbb{Z}^d)\leq Cn^{-1}.$$

The conjecture predicts a transition from exponential to stretched-exponential, and eventually polynomial behavior as the required separation grows; the source gives no resolution.