The polynomial growth conjecture for simple closed geodesics on nonorientable surfaces

Let $S$ be a nonorientable surface, let $m$ be a hyperbolic metric on $S$, and let $\gamma_0$ be a simple closed geodesic. Write $\mathcal{S}_{\gamma_0}$ for the mapping-class-group orbit of $\gamma_0$, and let $\delta$ be the Hausdorff dimension of $\mathcal{ML}^+(S)$. Denote the length of a geodesic $\gamma$ in the metric $m$ by $\ell_m(\gamma)$.

Simple-closed-geodesic growth conjecture. There exists a constant $c>0$ such that

\left|\left\\{\gamma\in\mathcal{S}_{\gamma_0}:\ell_m(\gamma)\leq L\right\\}\right|\simeq cL^\delta

as $L$ tends to infinity.

The conjecture is known when the Euler characteristic of $S$ is $-1$ and is stated to appear valid for the thrice-punctured projective plane. It remains open in the generality stated.