Collapsed pseudo-locality conjecture with scalar-curvature control

Let $\alpha\in(0,1)$. There should exist positive constants $\delta=\delta(m,\alpha)$ and $\varepsilon=\varepsilon(m,\alpha)$ such that, if $(M,g(t))_{t\in[0,T]}$ is an $m$-dimensional Ricci flow with complete time slices and, for some $p\in M$,

$$\sup_{B_{g(0)}(p,1)}|\operatorname{Rm}_{g(0)}|_{g(0)}\leq 1,\qquad d_{\rm GH}\bigl(B_{g(0)}(p,1),\mathbb B^k(1)\bigr)\leq\delta,\qquad \sup_{M\times[0,T]}|\operatorname{Sc}_{g(t)}|\leq 1,$$

then for every $t\in[0,\varepsilon^2]$,

$$\sup_{B_{g(t)}(p,\varepsilon)}|\operatorname{Rm}_{g(t)}|_{g(t)}\leq \alpha t^{-1}+\varepsilon^{-2}.$$

Collapsed pseudo-locality conjecture. The asserted curvature estimate should hold even when the initial unit ball is Gromov--Hausdorff close to a lower-dimensional Euclidean ball. This would provide a pseudo-locality principle compatible with collapse, beyond the usual noncollapsing hypotheses; examples cited in the source show that simply deleting all local volume assumptions without scalar-curvature control cannot work.