The asymptotic volume growth conjecture under positive scalar curvature

Let $(M^n,g)$ be a complete non-compact Riemannian manifold with Ricci curvature $Rc\geq 0$. Denote by $B_p(r)$ the open geodesic ball centered at $p\in M^n$ with radius $r>0$, and by $V(B_p(r))$ its volume. Let $R$ denote the scalar curvature.

Asymptotic volume growth conjecture. There is a universal constant $C(n)>0$ such that, if $R\geq 2$, then

$$\varlimsup_{r\rightarrow\infty}\frac{V(B_p(r))}{r^{n-2}}\leq C(n),\qquad\forall p\in M^n.$$

This is an asymptotic version of Gromov's volume conjecture, asserting an $r^{n-2}$ upper bound only at infinity rather than uniformly over all radii and centers. The source presents it as a conjecture and gives no resolution status.