Elashvili's conjecture on indices of centralizers

Let $\mathfrak{g}$ be a semisimple Lie algebra over an algebraically closed field of characteristic $0$, and let $x\in\mathfrak{g}$. By $C_{\mathfrak{g}}(x)$ denote the centralizer of $x$ in $\mathfrak{g}$, and by $\operatorname{ind}(K)$ the index of a finite-dimensional Lie algebra $K$. Elashvili's conjecture.

$$\operatorname{ind}(C_{\mathfrak{g}}(x))=\operatorname{rank}(\mathfrak{g}).$$

This conjecture concerns the index of centralizers of elements in semisimple Lie algebras. The paper proves it for the exceptional types by computer calculations; the statement is presented in the stated generality for semisimple Lie algebras.