When is the angular metric on the space of directions intrinsic?

Question

Suppose we have a point $p$ in an Alexandrov space $X$ of curvature bounded below and let $\Sigma_pX$ denote the space of directions of $X$ at $p$. What conditions on $X$ are necessary to ensure that $\Sigma_pX$ is an intrinsic (or inner) metric space—i.e., to ensure that the usual angular metric on $\Sigma_pX$ is intrinsic? As noted in Halbeisen's paper below, it is sufficient to require $X$ to be finite-dimensional, complete, and intrinsic itself. Can any of these conditions be weakened or dropped?

Halbeisen, Stephanie, On tangent cones of Alexandrov spaces with curvature bounded below, Manuscr. Math. 103, No. 2, 169-182 (2000). ZBL1021.53040.

Answer 1

One may ask if locally compact Alexandrov spaces have intrinsic spaces of directions --- I do not know the answer. See 13.40 in our book.