Yau's scalar-curvature growth conjecture for three-manifolds

Let $(M^3,g)$ be a three-dimensional complete manifold with Ricci curvature $\operatorname{Ric}\geq 0$, and let $B_p(R)$ denote the geodesic ball of radius $R$ centered at $p\in M$. Let $S$ be the scalar curvature of $M$. Yau's conjecture. The scalar curvature satisfies

$$\limsup_{R\rightarrow\infty}\frac{1}{R}\int_{B_p(R)}S\leq 8\pi.$$

This is the three-dimensional, sharp-bound version of Yau's question on the growth of the integral of scalar curvature on complete manifolds with nonnegative Ricci curvature. The bound is proved in the source when $S$ is bounded between two positive constants, and is also known when $M$ admits a pole; the conjecture without those restrictions remains open.