The $(T,\mathcal{V})$-proarrow conjecture for enriched approaches to automata

Let $T$ be a monad on $\operatorname{Set}$ and let $\mathcal{V}$ be a quantale. Write $T\mathcal{V}\operatorname{Prof}$ for the locally thin bicategory associated with $(T,\mathcal{V})$, whose varying instances yield categories of topological spaces, approach spaces, metric and ultrametric spaces, and closure spaces. The $(T,\mathcal{V})$-proarrow conjecture. When the reachability result referenced as $\autoref{reach_eq}$ is instantiated in $T\mathcal{V}\operatorname{Prof}$, it should provide a 2-categorical perspective on topological, metric, and, in a broad sense, fuzzy approaches to automata theory. This proposes a common 2-categorical framework for several generalized notions of space and distance, but the source does not state a precise equivalence or resolution.