The smoothness conjecture for invariant generically stable measures in n-distal groups

Assume $2\leq k\in\omega$, let $T$ be a $k$-distal NIP theory, and let $G$ be a definable group. Let $\mu\in\mathfrak{M}_{G}(\mathbb{M})$ be a generically stable, $G$-invariant measure. Smoothness conjecture. The measure $\mu$ is smooth. In particular, if $G$ is fsg, then $G$ is compactly dominated. More generally, the analogous question can be asked for fim groups in not necessarily NIP $k$-distal theories. Smoothness of such measures would extend the known distal result and yield compact domination and its arithmetic regularity consequences in the broader $k$-distal setting; the source does not indicate a resolution.