Extension of the parabolic Calogero–Moser results to all compositions

Let $\alpha=(\alpha_1,\dotsc,\alpha_{\ell})$ be a composition, and let $C_n$ denote the parabolic Calogero–Moser variety associated with $\alpha$. The results established under the hypothesis $\ell\leq 5$ include the complete-intersection theorem for the partial Grothendieck–Springer resolution, the description of irreducible components, and the smoothness, irreducibility, and dimension theorem for $C_n$.

Extension conjecture. Theorem~, Theorem~, and Theorem~ hold for all compositions $\alpha$.

This conjecture asserts that the restriction to at most five blocks is only a limitation of the methods used, particularly the reliance on the cited block-vector theorem. It would extend Nevins's conjecture from the considered parabolic subgroups to all parabolic subgroups.