Soul bundle conjecture for topologically nice Alexandrov spaces

Let $X$ be a complete open Alexandrov space with nonnegative curvature, and suppose that $X$ is topologically nice. Let $S$ be a soul of $X$, and let $B(S,\epsilon)$ denote its closed $\epsilon$-neighborhood.

Soul bundle conjecture. There exists a positive number $\epsilon$ such that

$$X\cong \operatorname{int} B(S,\epsilon),$$

and $B(S,\epsilon)$ is homeomorphic to a disk bundle over $S$, called the normal bundle of $S$.

The conjecture is stated as known in dimension three and is asserted by the source to be proved in dimension four. The supplied candidate does not include the definition of “topologically nice,” so that hypothesis should be checked against the paper.