Elashvili's index conjecture for centralizers of reductive Lie algebras

Let ${\mathfrak g}$ be a reductive Lie algebra, and let ${\mathfrak g}^x$ denote the centralizer of $x\in{\mathfrak g}^*$. Write ${\rm ind}\,{\mathfrak h}$ for the index of a Lie algebra ${\mathfrak h}$ and ${\rm rk}\,{\mathfrak g}$ for the rank of ${\mathfrak g}$. Elashvili's conjecture.

$${\rm ind}\,{\mathfrak g}^x={\rm rk}\,{\mathfrak g}\quad\text{for all }x\in{\mathfrak g}^*.$$

The conjecture predicts that every coadjoint centralizer of a reductive Lie algebra has index equal to the rank of the original algebra. It is motivated by the argument shift method and its relation to algebraically independent Poisson-commuting functions on coadjoint orbits; the supplied text does not state whether the conjecture has been resolved.