The Margulis-type cut-off covering spectrum conjecture

Let $M^n$ be a complete Riemannian manifold with $\operatorname{Ricci}\geq -1$, or a Gromov–Hausdorff limit of such spaces. Let $\operatorname{CovSpec}_{\mathrm{cut}}^{\rho}(M,p)$ denote the cut-off covering spectrum at scale $\rho$ based at $p$, and let a subscaled soul mean the soul structure referred to in the source near the relevant preceding definition. For all $b>a>0$, there should be a scale $\rho=\rho(a,b,n)$ with the following alternative.

The Margulis-type cut-off covering spectrum conjecture.

$$\operatorname{CovSpec}_{\mathrm{cut}}^{\rho}(M,p)\cap [a\rho,b\rho]=\emptyset$$

or there is a subscaled soul as in the source near $p$.

This is intended as a Margulis-lemma-type statement for manifolds with Ricci curvature bounded below and their Gromov–Hausdorff limits. It is presented as a prospective application of the preceding conjecture, and its resolution is not supplied in the source context.