Demailly's convergence conjecture for Monge–Ampère masses

Let $\varphi$ be plurisubharmonic in a bounded pseudoconvex neighborhood of $0\in\mathbb{C}^n$, locally bounded away from $0$. For $m\in\mathbb{Z}_{>0}$, let $\varphi_m$ be the analytic approximation defined from an orthonormal basis of the multiplier ideal $\mathcal{J}(m\varphi)$, and let $e_n(\varphi)=\int_{\{0\}}(\operatorname{dd^c}\varphi)^n$ denote the residual Monge–Ampère mass at $0$. Demailly's conjecture. As $m\rightarrow +\infty$, $e_n(\varphi_m)$ converges to $e_n(\varphi)$. This asks whether Demailly's analytic approximations preserve residual Monge–Ampère mass in the limit; the statement is presented as a conjecture in the source, and no resolution is supplied here.