The cut-off covering spectrum theorem for Ricci-limit spaces

Let $X$ be a Gromov–Hausdorff limit of complete noncompact Riemannian manifolds $M_i$ with $\operatorname{Ricci}(M_i)\geq -\epsilon_i$ and $\epsilon_i\to 0$. Let $\operatorname{CovSpec}_{\mathrm{cut}}(X)$ denote the cut-off covering spectrum, and let the preceding theorem assert that for the spaces under consideration either this spectrum is empty or the universal cover splits isometrically.

The cut-off covering spectrum conjecture. Theorem 7, namely the corresponding cut-off covering spectrum conclusion established earlier in the paper, holds for such limit spaces $X$.

The source presents this as a consequence of the preceding flat normal bundle conjecture: if that conjecture holds, the proof of the earlier theorem extends to the Ricci-limit setting. The precise theorem referred to by the internal label is not reproduced in the supplied context, so the mathematical scope of this row should be checked against the paper.