Weak-to-strong convergence conjecture for finite-energy metrics

Let $(X,L)$ be a polarized variety. Every $d_2$-bounded weakly convergent sequence in $\mathcal{E}_{\mathrm{NA}}^2(X,L)$ should be $d_1$-convergent. This conjecture would provide the missing convergence needed in the compactness approach to the non-archimedean Calabi-energy maximization problem; it remains open.