Uniform index conjecture under Euclidean volume growth

Given $n$ and $L\in(0,1]$, let $M$ be an open $n$-manifold with $\operatorname{Ric}\ge 0$, and suppose that the Riemannian universal cover of $M$ has Euclidean volume growth of constant at least $L$. Uniform index conjecture. The group $\pi_1(M)$ is finitely generated and contains an abelian subgroup of index at most $C(n,L)$. If true, this would strengthen the known virtual abelianness result for manifolds whose universal covers have Euclidean volume growth; the surrounding text indicates that it would follow from an affirmative answer to the cited question about the escape rate.