Non-Archimedean regularization conjecture for entropy and energy

Let $X_{NA}$ be the non-Archimedean analytification, let $\mathcal{E}^{1}(X_{NA})$ be the finite-energy space, let $\mathcal{H}(X_{NA})$ be the space of non-Archimedean model potentials, and let $MA$ denote the non-Archimedean Monge–Ampère operator. For $u\in\mathcal{E}^{1}(X_{NA})$, write $E_{NA}(MA(u))$ and $\operatorname{Ent}_{NA}(MA(u))$ for its non-Archimedean energy and entropy. Non-Archimedean regularization conjecture. Given any $u\in\mathcal{E}^{1}(X_{NA})$, there exists a sequence $u_j\in\mathcal{H}(X_{NA})$ converging weakly to $u$ such that

$$E_{NA}(MA(u_j))\to E_{NA}(MA(u)),\qquad \operatorname{Ent}_{NA}(MA(u_j))\to \operatorname{Ent}_{NA}(MA(u)).$$

Such a regularization would provide the non-Archimedean analogue of the regularization property used in the variational Yau–Tian–Donaldson argument; the source presents it as a statement to be proved and gives no resolution.