Volume growth and asymptotic cone dimension conjecture

Let $M^n$ be an open manifold with $\operatorname{Ric}_M\geq 0$ and

$$\inf_{x\in M}\operatorname{vol} B_1(x)>0.$$

Volume growth and asymptotic cone dimension conjecture. (1) If

$$\liminf_{R\to\infty} \frac{\operatorname{vol}(B_R(p))}{R^k}=0,$$

then there exists an asymptotic cone of $M$ with dimension $<k$. (2) If

$$\limsup_{R\to\infty} \frac{\operatorname{vol}(B_R(p))}{R^k}=0,$$

then every asymptotic cone of $M$ has dimension $<k$.

This conjecture seeks to relate subpolynomial volume growth along some or all scales to lower-dimensional asymptotic cones. The source presents it as motivated by the preceding theorem and corollary; no resolution is given.