The limit-set conjecture for the mapping class group of a nonorientable surface

Let $S$ be a nonorientable hyperbolic surface. The limit set of the mapping class group $\operatorname{Map}(S)$ in $\mathcal{PML}(S)$ is the set $\mathcal{PML}^+(S)$ of projective measured laminations without one-sided closed leaves.

Limit-set conjecture. The limit set of $\operatorname{Map}(S)$ in $\mathcal{PML}(S)$ is $\mathcal{PML}^+(S)$.

The paper states that this conjecture is proved for genus-one surfaces; its status in general is not resolved here.