The fiber-bundle conjecture for Sharafutdinov projections

Let $X\in \operatorname{Alex}^n(0)$ be an open Alexandrov space, let $S$ be a compact totally convex soul without boundary, and let $\phi:X\to S$ be the Sharafutdinov projection. Write $m=\dim(S)\geq 1$. A point is weakly $k$-strained if it admits the configuration of $(k+1)$ points used to define weak straining in the source, with the relevant comparison angles greater than $\pi/2$.

Fiber-bundle conjecture. If all points in $S$ are weakly $m$-strained, then $\phi:X\to S$ is a fiber bundle map.

The conjecture proposes sufficient regularity of the soul for the Sharafutdinov projection to be a fiber bundle. The source emphasizes that submetry alone does not imply the fiber-bundle property and does not state a resolution.