Regular-sequence conjecture for Mishchenko–Fomenko algebras of nilpotent centralizers

Let $\mathfrak{g}$ be a reductive Lie algebra, let $e$ be a nilpotent element, and write $\mathfrak{g}^{e}$ for its centralizer. Let $\mathcal{F}_\xi(\mathfrak{g}^{e})$ be the Mishchenko–Fomenko algebra associated with $\xi\in(\mathfrak{g}^{e})^*$, and let $(\mathfrak{g}^{e})^*_{{\rm reg}}$ denote the regular elements. Assume that condition $(\ast)$ and condition $(2)$ of Theorem~ hold. Regular-sequence conjecture. The free generators of $\mathcal{F}_\xi(\mathfrak{g}^{e})$ form a regular sequence for every $\xi\in(\mathfrak{g}^{e})^*_{{\rm reg}}$. The claim extends the known regular-sequence results for invariant algebras and the cases where the relevant hypotheses and quantization results are already established; the general assertion is left open in the source.