C^0 lower semi-continuity conjecture for entropy of surface metrics

Let $S$ be a closed orientable surface, and let $\operatorname{Met}(S)$ denote the space of Riemannian metrics on $S$. Consider the entropy functional

$$h_{\rm top}:\operatorname{Met}(S)\to[0,\infty).$$

Surface-metric entropy stability conjecture. The functional $h_{\rm top}$ is lower semi-continuous with respect to $\overline{d}_{C^0}$.

This conjecture concerns $C^0$-stability of topological entropy for metrics on closed orientable surfaces and is presented among the paper's proposed future directions. Its resolution is not specified in the supplied text.