On dimensions of tangent cones in limit spaces with lower Ricci curvature bounds

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.