Codimension-three regularity of noncollapsed Ricci limit spaces under an integral volume-deficit bound

Let $(X,d,p)$ be a noncollapsed pointed Gromov–Hausdorff limit of complete $n$-dimensional Riemannian manifolds with a uniform lower Ricci curvature bound, where $n\ge 4$. We assume that, on each bounded ball, the integral of the $3\mathrm{/}2$ power of the small-ball volume deficit relative to the hyperbolic comparison volume is $O(r^3)$ as $r\to 0$. We prove that the metric singular set has Hausdorff dimension at most $n-3$ and sigma-finite $(n-3)$-dimensional Hausdorff measure, thus confirming a particular case of codimension-three regularity conjecture \cite[Conjecture 2.4]{Naber2020}.