Integral scalar-curvature formula for three-manifolds with nonnegative Ricci curvature

Let $(M^3,g)$ be a complete non-compact Riemannian manifold, let $q\in M$, and suppose that $Rc\geq 0$ and $M$ has maximum volume growth. Writing $R$ for the scalar curvature and $\mathrm{V}_M$ for the asymptotic volume ratio, the integral scalar-curvature formula.

$$\lim_{r\rightarrow\infty}\frac{\int_{B_q(r)}R}{r}=8\pi\big[1-\mathrm{V}_M\big].$$

The conjecture concerns the asymptotic total scalar curvature of three-manifolds with nonnegative Ricci curvature and maximal volume growth. It was proved for manifolds with a pole, while the general case remains open.