Break the n^{2/3}+D upper bound for the reachability version of replacement paths

Develop an algorithm in the CONGEST model that improves upon the current \widetilde{O}(n^{2/3} + D) round upper bound for the reachability version of the Replacement Paths problem, which asks, for each edge e on a given s–t shortest path P, whether s can reach t in G \ e.

Background

Beyond distance approximation, the authors consider a simpler reachability variant—deciding for each edge in the given s–t path whether connectivity is preserved when that edge is removed.

Despite being ostensibly easier than computing distances, the current best upper bound remains \widetilde{O}(n^{2/3} + D), and the authors explicitly note that they do not know how to surpass this bound.