Conjectural two-sided bounds for harmonic measure in the Liouville quantum gravity setting

Let $\widehat{q}(y)$ be the quantity defined in the paper from the harmonic measure of the embedded graph, and let $y$ tend to infinity in the Euclidean embedding. Harmonic-measure bounds conjecture. There exist nonrandom constants $a,b>0$ such that, almost surely,

$$a=\liminf_{y\to\infty}\widehat{q}(y)\leq\limsup_{y\to\infty}\widehat{q}(y)=1-b.$$

The preceding theorem gives only a one-sided upper bound and the universal lower bound $1/2$ for the limsup. The conjecture predicts strictly positive gaps from both $0$ and $1$; the paper discusses possible sharp values from extremal volume exponents and expects analogous behavior for models such as the UIPT.