Path connectedness of the regular part of non-collapsed RCD spaces

Let $(X,\mathsf d,\mathcal{H}^{N})$ be a non-collapsed $\mathsf{RCD}(K,N)$ space, and let $({\mathcal R}_{N})_{\varepsilon}$ denote its $\varepsilon$-regular set. Then the interior $\overset{\circ}{({\mathcal R}_{N})_{\varepsilon}}$ is path connected for all sufficiently small $\varepsilon$. Path-connectedness conjecture. The set

$$\overset{\circ}{({\mathcal R}_{N})_{\varepsilon}}$$

is path connected for all small $\varepsilon$. The authors suspect that the hypothesis $\mathcal{H}^{N-1}(\mathcal{S})=0$ used in the corresponding theorem is unnecessary; this conjecture asserts the resulting statement for every non-collapsed $\mathsf{RCD}(K,N)$ space.