Demailly's strong continuity conjecture for Monge–Ampère masses

Let $D$ be an open ball in $\mathbf{C}^n$, and let $\varphi$ be a plurisubharmonic function with an isolated singularity at $0\in D$. Let $\{\varphi_m\}_{m\geq 1}$ be the Demailly approximation sequence of $\varphi$.

Demailly's conjecture. The $n$-th Lelong numbers converge:

$$L_n(\varphi_m,0)\to L_n(\varphi,0)\quad\text{as }m\to\infty.$$

This is a strong continuity statement for Monge–Ampère operators under Demailly approximation. The paper notes that the result is known for Green and Siu functions associated to graded systems of $\mathfrak{m}$-primary ideals, while the general case remains open in the supplied context.