Finite entropy and non-Archimedean energy conjecture

Let $(X,L)$ be a polarized manifold, let $\mathcal{R}^1$ be the space of test curves, and let $\mathcal{E}^{1,\mathrm{NA}}(L)$ be the subspace consisting of those $ell\in\mathcal{R}^1$ whose Legendre transform $hat{\ell}_{\tau}$ is either $-\infty$ or $\mathscr{I}$-model for every $\tau$. Let $\mathbf{Ent}(\ell)$ denote the entropy of a test curve. Finite entropy conjecture. If $ell\in\mathcal{R}^1$ and $\mathbf{Ent}(\ell)<\infty$, then $ell\in\mathcal{E}^{1,\mathrm{NA}}(L)$. The conjecture is known when $[\omega]$ is integral, by the cited work of Li, but remains open in the general setting described here.