Lipnowski–Wright's conjecture on simple and non-simple geodesics

Let $\mathcal{M}_g$ be the moduli space of closed hyperbolic surfaces of genus $g$. For a surface $X_g$ and $L>0$, let $N(X_g,L)$ count unoriented primitive closed geodesics of length at most $L$, and let $N^s_{nsep}(X_g,L)$ count those that are simple and non-separating; call a geodesic non-simple if it is not simple.

Lipnowski–Wright's conjecture. As $g\to\infty$, on most closed hyperbolic surfaces in $\mathcal{M}_g$, most geodesics of length significantly less than $\sqrt g$ are simple and non-separating, whereas most geodesics of length significantly greater than $\sqrt g$ are non-simple.

The conjecture concerns the transition near the scale $\sqrt g$ between simple non-separating and non-simple closed geodesics on Weil–Petersson-generic large-genus surfaces. The paper's abstract says that its results confirm this conjecture, so its status is recorded as solved.