Period–Index Conjecture for Function Fields

Let $X$ be a $d$-dimensional irreducible variety over an algebraically closed field $k$, and let $k(X)$ denote its function field. For $\alpha\in\operatorname{Br}(k(X))$, write $\operatorname{per}(\alpha)$ and $\operatorname{ind}(\alpha)$ for its period and index.

Period–Index Conjecture for Function Fields. For every $\alpha\in\operatorname{Br}(k(X))$,

$$\operatorname{ind}(\alpha)\mid\operatorname{per}(\alpha)^{d-1}.$$

This is described as one of the major outstanding conjectures in the study of division algebras; the supplied source gives no evidence that it has been resolved.