Local compactness conjecture for convex functionals on surfaces

Let $S$ be a surface, and let $\operatorname{Convex}(S)$ denote the space of convex curve functionals on $S$, equipped with the product topology inherited from its values on the curve set $\mathcal{C}(S)$. Local compactness conjecture for $\operatorname{Convex}(S)$. The set $\operatorname{Convex}(S)$ with the product topology is locally compact. This is suggested by the local compactness of the corresponding set $\operatorname{AConvex}(S)$ of additive convex functionals and by the compact-convex-set model in train-track coordinates; the supplied text does not state that the conjecture has been resolved.