Minimax conjecture for Calabi energy

Suppose $(X,L)$ has only klt singularities. The non-archimedean Calabi energy should satisfy

$$\sup_{\varphi\in\mathcal{E}_{\mathrm{NA}}^2(X,L)}C_{\mathrm{NA}}(\varphi)=\inf_{\omega_\phi\in\mathcal{E}^2(X,L)}C(\omega_\phi),$$

with an appropriate definition of the right-hand side. This would identify the non-archimedean maximization problem with the analytic minimization problem; the equality is open in general.