The codimension-three conjecture for the topologically singular set of RCD spaces

Let $(X,\mathsf d,\mathcal{H}^{N})$ be a non-collapsed $\mathsf{RCD}(K,N)$ space. Denote by ${\mathcal R}_{N}$ its $N$-regular set. Codimension-three conjecture. There exists an open subset $M\subset X$ such that $M$ is homeomorphic to a topological manifold, ${\mathcal R}_{N}\subset M$, and $X\setminus(\boldsymbol\partial X\cup M)$ has codimension at least $3$. This would sharpen the known decomposition by improving the codimension bound for the topologically singular set from $2$ to $3$; the bound is sharp in general because the vertex of $C(\mathbb{RP}^{2})$ is not a manifold point.