The density conjecture for rational projective measured laminations

Let $S$ be a nonorientable surface with Euler characteristic $\chi(S)<-1$. Let $\mathcal{PML}^+(S)$ be the projective measured laminations without one-sided closed leaves, and let $\mathcal{PML}^+(S,\mathbf{Q})$ denote the rational points in this space.

Rational-density conjecture.

$$\mathcal{PML}^+(S)=\overline{\mathcal{PML}^+(S,\mathbf{Q})}.$$

The equality is known for genus-one surfaces according to the source. The conjecture is proposed as a first step toward the broader closure equalities stated subsequently.