Stability and integrality of volume growth order for manifolds with noncollapsed universal cover

Determine whether, for every open manifold M with nonnegative Ricci curvature and noncollapsed Riemannian universal cover, the infimum and supremum of volume growth order coincide and are positive integers; specifically, ascertain whether IV(M)=SV(M)∈N+.

Background

The paper defines IV(M) and SV(M) as the infimum and supremum of the volume growth order of an open manifold M with nonnegative Ricci curvature. Under the hypothesis IV(M)<2 and noncollapsed universal cover, Theorem A (Theorem noncollapserigidity) shows a rigidity that implies IV(M)=SV(M)=1 in this case.

Motivated by this result, the author asks whether the equality and integrality of IV(M) and SV(M) hold more generally for all such manifolds with noncollapsed universal cover, not only under the condition IV(M)<2.