Lu's idempotence conjecture for the singularity envelope

Let $(X,\omega)$ be a compact Kähler manifold, let $\theta$ be a smooth closed real $(1,1)$-form representing a big class, and let $P_\theta[u]$ denote the singularity envelope of $u\in\operatorname{PSH}(X,\theta)$. Lu's idempotence conjecture. For every $u\in\operatorname{PSH}(X,\theta)$,

$$P_\theta[P_\theta[u]]=P_\theta[u].$$

This asserts that applying the singularity-envelope operator twice has the same effect as applying it once. The supplied text gives no further information about known cases or resolution, so the conjecture remains open here.