The soul conjecture for open Alexandrov spaces

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.

Soul conjecture. If $X$ contains an open subset where the curvature is bounded below by a positive constant, then $S$ is a point.

This is a conjectured Alexandrov-geometric counterpart of the Riemannian soul theorem. The source notes that the submetry conjecture implies this conjecture, but gives no resolution.