Liouville-component conjecture for semiclassical measures

Let $(M,g)$ be a compact manifold with Anosov geodesic flow, let $\mu$ be a semiclassical measure on $S^*M$, and let $\mu_L$ denote the Liouville measure. Liouville-component conjecture. There exist $\alpha\in(0,1]$ and a probability measure $\mu'$ on $S^*M$ such that

$$\mu=\alpha\mu_L+(1-\alpha)\mu'.$$

This conjecture would provide a positive Liouville component for every semiclassical measure and would substantially advance Quantum Unique Ergodicity, although it would not exclude the quantum-cat-map counterexample discussed in the source.