If $X$ is an Alexandrov space of curvature bounded below by a real number $k$, is it true that any geodesic in the tangent cone $T_pX$ can be realized as a limit of geodesics when we view $T_pX$ as the Gromov-Hausdorff limit of rescalings of neighborhoods of $p$?

More generally, if $X_i$ is a sequence of Alexandrov spaces with curvatures bounded below uniformly by $k$ and $\dim X_i=n$, such that $X_i$ converges in the Gromov-Hausdorff sense to an Alexandrov space $X$ with $\dim X=n$ (i.e., the sequence is non-collapsing), is it true that any geodesic in $X$ is the limit of a sequence of geodesics in $X_i$?