Conjecture on the growth constant for harmonic measures on $\mathbb{Z}^2$

Let $\psi(\mathbb{Z}^2)$ denote the reciprocal extremal factor governing the exponential lower bound for positive harmonic measures on finite subsets of $\mathbb{Z}^2$. The proposed extremal sets $D_n$ satisfy $\mathbb{H}_{D_n}((0,0))=(2+\sqrt{3})^{-2n+o(n)}$. Growth-constant conjecture.

$$\psi(\mathbb{Z}^2)=(2+\sqrt{3})^2.$$

If true, this gives the exact exponential rate associated with the least positive harmonic measures; the source says that proving this equality would suffice for the preceding extremal-value statement, but does not report a resolution.