Codimension-5 conjecture for noncollapsed Einstein 5-manifold limits

Let $(X,d)$ be a Gromov–Hausdorff limit of noncollapsed Einstein $5$-manifolds. Let $\mathcal{R}_{\mathrm{orb}}$ be the orbifold regular set and define the orbifold singular set by

$$\mathcal{S}_{\mathrm{orb}}=X\setminus\mathcal{R}_{\mathrm{orb}}.$$

Codimension-5 conjecture. The set $\mathcal{S}_{\mathrm{orb}}$ is countable. In particular, away from a countable set of points, $(X,d)$ has the structure of a smooth Einstein $5$-orbifold with singularities of the form $\mathbb{R}\times\mathbb{R}^4/\Gamma$ along countably many geodesics. This conjecture would follow essentially from ruling out accumulation of singular points not lying on curves of the singular set, and it proposes an orbifold regularity theory off a codimension-$5$ set.