Noncollapsed-cover quantitative space-form rigidity conjecture

Let $(M,g)\in\mathcal{M}_{\rm Rc}(m)$, and let $(\widetilde M,\widetilde g)$ be its Riemannian universal covering. The quantitative space-form rigidity theorem concerns lower bounds on the local rewinding volume and, depending on the value of the Ricci lower-bound parameter, concludes that $M$ is respectively a spherical space form, a flat manifold, or a hyperbolic manifold. Noncollapsed-cover rigidity conjecture. The same theorem should remain valid without assuming that there exists $\widetilde p_0\in\widetilde M$ such that

$$|B_{\widetilde g}(\widetilde p_0,1)|_{\widetilde g}\geq v>0.$$

Removing this universal-cover noncollapsing hypothesis would extend the quantitative rigidity result to potentially collapsed universal covers; the source presents this as a question rather than a resolved theorem.