Higher codimension area-minimizing currents mod$(q)$: structure of singularities near $(m-1)$-invariant cones

We study finer properties related to the interior regularity of m-dimensional area minimizing currents mod$(q)$ in arbitrary codimension. We show that the set of points where at least one tangent cone is translation invariant along $m-1$ directions is locally a connected C1,β submanifold, and moreover such points have unique tangent cones. We establish these results as consequences of a fine excess decay theorem.