Robustness conjecture for positive topological entropy on closed surfaces

Let $Q$ be a closed surface, let $g$ be a Riemannian metric on $Q$, and write $h_{\rm top}(g)$ for the topological entropy of its geodesic flow. Say that $h_{\rm top}(g)$ is robust if it remains positive for all metrics sufficiently close to $g$ in the $C^0$ topology. Robustness conjecture. If $Q$ is a closed surface, then $h_{\rm top}(g)$ is robust whenever it does not vanish. The paper establishes robust positive topological entropy for generic metrics on the 2-torus and density of robust high-entropy metrics in every dimension at least two; the conjecture proposes the corresponding nonvanishing criterion for all closed surfaces.