  We show that if $X$ is a limit of $n$-dimensional Riemannian manifolds with
Ricci curvature bounded below and $\gamma$ is a limit geodesic in $X$ then
along the interior of $\gamma$ same scale measure metric tangent cones
$T_{\gamma(t)}X$ are H\"older continuous with respect to measured
Gromov-Hausdorff topology and have the same dimension in the sense of
Colding-Naber.
