$*$-connectivity conjecture for extremal sets in $\mathbb{Z}^d$

For $d\geq3$ and $n\geq2$, let $\mathfs{M}_n(\mathbb{Z}^d)$ be the least positive harmonic measure among subsets of $\mathbb{Z}^d$ of cardinality $n$. Two lattice vertices are $*$-adjacent when they are adjacent under the $*$-connectivity notion defined in the source. $*$-connectivity conjecture. For every integer $n\geq2$, the sets achieving $\mathfs{M}_n(\mathbb{Z}^d)$ are $*$-connected. The conjecture is motivated by transience in dimensions $d\geq3$, which makes very sparse sets ineffective at producing exponential decay; the source gives no resolution.