The Lipschitz-rate and constant-density conjecture for collapsed Ricci limits

Let $\gamma\colon(-2,2)\to X$ be a minimizing geodesic, and let $s,t\in(-1,1)$.

Volume-density conjecture. We have

$$\frac{\nu(B_r(\gamma(s)))}{\nu(B_r(\gamma(t)))}\leq C(n,\lambda)|t-s|,$$

with the Lipschitz-rate interpretation, and, if $X$ is noncollapsed,

$$\lim_{r\to0}\frac{\nu(B_r(\gamma(s)))}{\nu(B_r(\gamma(t)))}=1.$$

The known Hölder variation rate for geometries along geodesics is sharp, and the conjecture predicts stronger behavior for volume ratios. The source gives no resolution.