Quantum unique ergodicity conjecture for Liouville quantum gravity

Fix $\gamma\in(0,2)$, let $\mu_\gamma$ be the Liouville measure on a bounded domain $D$, and let $(\mathbf{f}_n)$ be eigenfunctions normalised to have unit $L^2(\mu_\gamma)$ norm. Liouville quantum unique ergodicity conjecture. As $n\to\infty$,

$$|\mathbf{f}_n(x)|^2\mu_\gamma(dx)\Rightarrow\frac{\mu_\gamma(dx)}{\mu_\gamma(D)}$$

in the weak-$*$ topology in probability. This extends quantum unique ergodicity predictions from negatively curved deterministic surfaces to the Liouville conformal field theory setting. The source gives no resolution.