PL-structure conjecture for Alexandrov spaces without proper extremal subsets

Let $X$ be an Alexandrov space, meaning a finite-dimensional complete length space with curvature bounded below in the Alexandrov sense, and suppose that $X$ contains no proper extremal subsets. An iterated space of directions is obtained by repeatedly taking the space of directions at a point. PL-structure conjecture. Every iterated space of directions of $X$ is homeomorphic to a sphere. This conjecture is motivated by the theorem that every space of directions of such an Alexandrov space is homeomorphic to a sphere; the stronger assertion is intended to imply that these spaces admit some kind of PL structure, but its general validity remains open.