Square-spiral conjecture for the least positive harmonic measure

Let $o=(0,0)$ and let $\mathscr{H}_n$ be the collection of $n$-element subsets $A$ of $\mathbb{Z}^2$ such that $\mathrm{H}_A(o)>0$. Here $\mathrm{H}_A(o)$ denotes the harmonic measure of $A$ at $o$.

Square-spiral conjecture. Asymptotically, the square spiral realizes the least positive value of harmonic measure, in the sense that

$$\lim_{n\to\infty}-\frac{1}{n}\log\inf_{A\in\mathscr{H}_n}\mathrm{H}_A(o)=2\log(2+\sqrt{3}).$$

The preceding theorem gives a general lower bound of order $e^{-cn\log n}$ and an improved bound of order $e^{-cn}$ for connected sets. The conjecture predicts the precise exponential decay rate, motivated by the square-spiral example and the associated tunnel heuristic; the source does not establish the claimed limit.