Archimedean/non-archimedean μ-entropy equality conjecture

Let $(X,L)$ be a polarized variety. Denote by $\mathcal{H}^{\mathbb{R}}_{\mathrm{NA}}(X,L)$ the space of non-archimedean metrics and by $\mathcal{H}(X,L)$ the space of Kähler metrics, with $\bm{\check{\mu}}^\lambda_{\mathrm{NA}}$ and $\bm{\check{\mu}}^\lambda$ the corresponding non-archimedean and Archimedean μ-entropies. μ-entropy equality conjecture. For $\lambda\in\mathbb{R}$, one has

$$\sup_{\phi\in\mathcal{H}^{\mathbb{R}}_{\mathrm{NA}}(X,L)}\bm{\check{\mu}}^\lambda_{\mathrm{NA}}(\phi)=\inf_{\omega_\varphi\in\mathcal{H}(X,L)}\bm{\check{\mu}}^\lambda(\omega_\varphi).$$

Moreover, when $\lambda\leq0$, $\bm{\check{\mu}}^\lambda_{\mathrm{NA}}$ should admit a maximizer in $\mathcal{H}^{\mathbb{R}}_{\mathrm{NA}}(X,L)$, unique modulo the action of $\operatorname{Aut}(X,L)$. The conjecture is presented as a far-reaching generalization of equality results relating canonical metrics and non-archimedean stability; the existence and uniqueness of the maximizer remain part of the conjectural statement.