Heat kernel geometry and Gromov's volume growth conjecture

In 1986, Gromov asked whether every complete noncompact $n$-dimensional Riemannian manifold with nonnegative Ricci curvature and scalar curvature at least one satisfies: $${\text{Vol}}_g(B(p,R))\le C_n{R}^{n-2}$$ for all $p\in M$ and $R>0$. We answer this question affirmatively using the heat-kernel Fisher metric and Nash entropy.