Boucksom–Jonsson's regularization conjecture

Let $(X,L)$ be a polarized manifold, let $\mathcal{E}^{1,\rm NA}(L)$ be the finite-energy non-Archimedean metrics, let ${\mathcal H}^{\rm NA}(L)$ be the smooth positive non-Archimedean metrics associated to semiample test configurations, and let ${\rm MA}^{\rm NA}$ denote the non-Archimedean Monge–Ampère measure. Let $A_X$ be the log discrepancy function on $X^{\rm NA}$. Regularization conjecture. For any $\phi\in \mathcal{E}^{1,\rm NA}(L)$, there exists a sequence $\{\phi_m\}\subset {\mathcal H}^{\rm NA}(L)$ converging to $\phi$ in the strong topology such that

$$\int_{X^{\rm NA}}A_X(v){\rm MA}^{\rm NA}(\phi)=\lim_{m\rightarrow +\infty}\int_{X^{\rm NA}}A_X(v){\rm MA}^{\rm NA}(\phi_m).$$

Boucksom and Jonsson introduced this conjecture in their non-Archimedean approach to K-stability. The source states that it is not known in general and that it would imply the entropy-slope conjectures and the uniform Yau–Tian–Donaldson conjecture in the setting considered there.