Yamaguchi's tangent-splitting conjecture for noncompact Alexandrov spaces

Let $A\in \operatorname{Alex}^n(0)$ be a noncompact Alexandrov space with a soul $S$. Let $C_i$ be the subsets associated with the soul as in 1.1, and let $p\in C_i\setminus\partial C_i$ be a topologically regular point. Let $T_pX$ denote the tangent cone at $p$, and let $K$ be a Euclidean cone. Yamaguchi's conjecture. The tangent cone $T_pX$ is isometric to the product

$$T_pC_i\times K.$$

This conjecture concerns the expected local product structure of nonnegatively curved Alexandrov spaces along the soul decomposition. The supplied source identifies it as a conjecture of Yamaguchi; no resolution is given here.