Scaling nonvanishing conjecture for isometries of noncollapsed Ricci-limit spaces

Let $(M,p)$ be a complete $n$-manifold satisfying

$$\operatorname{Ric}\ge -(n-1),\quad \operatorname{vol}(B_1(p))\ge v>0.$$

For an isometry $f$ of $M$, let

$$D_{r,p}(f)=\sup_{q\in B_r(p)}d(f(q),q).$$

The isometry $f$ is scaling $\Phi$-nonvanishing at $p$ if $s^{-1}D_{s,p}(f)\ge\delta>0$ for some $s\in(0,1]$ implies $r^{-1}D_{r,p}(f)\ge\Phi(\delta)$ for all $r\in(0,s].$

Scaling nonvanishing conjecture. Given $n$ and $v>0$, there is a positive function $\Phi(\delta,n,v)$ such that every isometry of $M$ is scaling $\Phi(\delta,n,v)$-nonvanishing at $p$.

The conjecture asserts that noncollapse of the unit ball forces a uniform link between the displacement of an isometry at different scales. If true, it would remove the scaling nonvanishing assumption from the paper's short-generator and finite-generation results.