The Sullivan-type invariant-measure conjecture for nonorientable surfaces

Let $S$ be a nonorientable surface, and let $\mathcal{ML}^+(S)$ be the cone of measured laminations without one-sided closed leaves. Let $\delta$ be the Hausdorff dimension of $\mathcal{ML}^+(S)$. A Radon measure is $\delta$-homogeneous if scaling by $t>0$ multiplies its measure by $t^\delta$.

Sullivan-type invariant-measure conjecture. There exists a unique, up to scaling, ergodic $\operatorname{Map}(S)$-invariant Radon measure supported on $\mathcal{ML}^+(S)$, and it is $\delta$-homogeneous.

This is presented as an analogue of Sullivan's theorem for geometrically finite Kleinian groups. The source gives no resolution of the conjecture.