The volume growth conjecture for Ricci-nonnegative manifolds

Let $(M^n,g)$ be a complete non-compact Riemannian manifold, let $B_R(p)$ denote the geodesic ball of radius $R$ centered at $p\in M$, and define

$$\Theta^{n-2,*}(\infty):=\limsup_{R\to\infty}\frac{\operatorname{vol}(B_R(p))}{R^{n-2}}.$$

Write $\operatorname{Ric}$ and $\operatorname{Sc}$ for the Ricci and scalar curvatures. The volume growth conjecture. There exists a constant $C(n)$ such that, if $\operatorname{Ric}\geqslant 0$ and $\operatorname{Sc}\geqslant 2$, then

$$\Theta^{n-2,*}(\infty)\leqslant C(n).$$

This is the asymptotic version of Gromov's volume bound question and concerns volume growth at infinity under nonnegative Ricci curvature and positive scalar curvature. The source makes the conjecture explicit by reference to earlier work, while the supplied text gives no resolution.