Lehmann's diminished-ideal containment conjecture

Let $X$ be a smooth variety, let $L$ be a pseudo-effective $\mathbb{R}$-divisor on $X$, and let $T_{\min}$ be a current of minimal singularities in the numerical class of $L$. Write $\mathcal{J}(T_{\min})$ for its multiplier ideal and $\mathcal{J}_{\sigma}(L)$ for the diminished ideal of $L$. Lehmann's diminished-ideal containment conjecture. One has

$$\mathcal{J}(T_{\min})\subset \mathcal{J}_{\sigma}(L).$$

This is the natural pseudo-effective analogue of the big-divisor equality conjecture. The text says that the containment can be strict in examples and that the author's main theorem proves it for most pseudo-effective divisors, but leaves the general case open.