Connected versus rectifiably connected support for uniqueness

Establish whether Theorem 1 (uniqueness up to an additive constant of Kantorovich potentials under rectifiable connectivity of supp ρ and bounded supp μ) remains valid when only assuming that the support of ρ is connected (without rectifiable connectivity).

Background

Theorem 1 requires rectifiable connectivity of supp ρ. The authors note that connectedness (e.g., path-connectedness by possibly infinite-length paths) may not guarantee the same regularity along curves used in their proof (which relies on Lipschitz parameterizations via rectifiable paths).

They highlight examples (e.g., graphs of Brownian paths) that are connected but not rectifiably connected, motivating the question whether mere connectedness suffices for uniqueness up to constants.