Euclidean volume growth conjecture for the escape rate

Let $(M,x)$ be an open manifold with $\operatorname{Ric}\ge 0$, let $\widetilde{M}$ be its Riemannian universal cover, and let $E(M,x)$ denote the escape rate. Euclidean volume growth conjecture. If $\widetilde{M}$ has Euclidean volume growth, then

$$E(M,x)=0.$$

The claim extends the preceding result for universal covers satisfying stronger Euclidean or stability-at-infinity hypotheses. Its general validity for universal covers with Euclidean volume growth remains open.