The splitting conjecture for integrable 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. Let $\pi:(\widetilde X,\widetilde p)\to(X,\bar p)$ be the metric universal cover, with $\bar p\in S$.

Splitting conjecture. If $\phi$ is integrable, then

$$\widetilde X=\widetilde S\times\phi^{-1}(\bar p),$$

where $\widetilde S=\pi^{-1}(S)$ splits as $\widehat S\times\mathbb R^k$ with $\widehat S$ compact.

This is the conjectured Alexandrov-geometric counterpart of the corresponding Riemannian rigidity theorem. The source does not state whether it is resolved.