Largest-loop exponent conjecture for uniform meandric systems

Let $\mathfrak S_n$ be a uniform meandric system of size $n$, and order its loops by decreasing size. For each fixed $k\in\mathbb N$, let the $k$th largest loop be measured by its number of vertices. Largest-loop exponent conjecture. With probability tending to $1$ as $n\to\infty$,

$$\#\{\text{vertices of the $k$th largest loop of $\mathfrak S_n$}\}=n^{\alpha+o(1)},\qquad \alpha=\frac12(3-\sqrt2)\approx0.7929.$$

This is a KPZ-based prediction for the macroscopic loop sizes in uniform meandric systems and remains unproved.