UIMS origin-loop tail exponent conjecture

Let $(\mathcal M,\Gamma)$ be the planar map and loop collection associated with the UIMS, and let $\ell_0$ be the loop or infinite path in $\Gamma$ passing through $0\in\mathbb Z$. Write $d$ for the exponent appearing in the paper's graph-distance bound and set $\alpha=\frac12(3-\sqrt2)$. UIMS origin-loop tail conjecture. As $k\to\infty$,

$$\mathbb P\mathopen{}\mathclose\bgroup\originalleft[\operatorname{diam}_{\mathcal M}(\ell_0)\ge k\aftergroup\egroup\originalright]\ge k^{-d(1-\alpha)-o(1)},$$

and

$$\mathbb P\mathopen{}\mathclose\bgroup\originalleft[\#\{\text{vertices in }\ell_0\}\ge k\aftergroup\egroup\originalright]\ge k^{-(1/\alpha-1)-o(1)}.$$

The paper states that the proven exponent is not expected to be optimal; these sharper lower-tail predictions remain open.