Regularization conjecture for non-archimedean entropy

Let $(X,L)$ be a polarized variety. For any $\varphi\in\mathcal{E}_{\mathrm{NA}}^1(X,L)$, there exists a sequence $\{\varphi_i\}_{i\in\mathbb{N}}\subset\mathcal{H}_{\mathrm{NA}}(X,L)$ converging to $\varphi$ in the strong topology such that

$$\lim_{i\to\infty}\int_{X^{\mathrm{NA}}}A_X\operatorname{MA}(\varphi_i)=\int_{X^{\mathrm{NA}}}A_X\operatorname{MA}(\varphi).$$

The conjecture is known for $T$-invariant metrics on toric varieties in the cited work and would extend entropy regularization from model metrics to finite-energy metrics; it remains open generally.