On the collapsing of spaces with local rewinding Reifenberg control

Let $(M^n,g_0)$ be a complete Riemannian manifold with $\Ric_{g_0}\ge-(n-1)g_0$ whose local universal covers are uniformly $δ$-Reifenberg-close to $\R^n$ below a fixed radius $ρ$. We show that for every $α\in(0,1)$ there are $δ(n,α)>0$ and $T(n,ρ,α)>0$ such that, if $δ\leδ(n,α)$, a complete Ricci fl