Panyushev–Premet–Yakimova conjecture on symmetric invariants of centralisers

Let $\mathfrak g$ be a finite-dimensional reductive Lie algebra of rank $l$ over an algebraically closed field of characteristic zero. For $x\in\mathfrak g$, let $\mathfrak g_x$ be its centraliser, and let ${\mathcal S}(\mathfrak g_x)^{\mathfrak g_x}$ denote the invariant subalgebra of the symmetric algebra of $\mathfrak g_x$ under the adjoint action. Panyushev–Premet–Yakimova conjecture. For every $x\in\mathfrak g$, the algebra

$${\mathcal S}(\mathfrak g_x)^{\mathfrak g_x}$$

is a graded polynomial algebra in $l$ variables. The conjecture is known for $x=0$, for regular nilpotent elements, and for all elements when $\mathfrak g$ is of type $\mathbf A$ or $\mathbf C$; the general case is the subject of the paper.