The unramified period-index conjecture for smooth projective varieties

Let $X$ be a smooth projective variety of dimension $d$ over an algebraically closed field $k$. For a Brauer class $\alpha\in\operatorname{Br}(X)$, define its period and index using its restriction to the generic point of $X$.

Unramified period-index conjecture. For every $\alpha \in \operatorname{Br}(X)$,

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

This is the unramified formulation of the period-index problem and, for fixed $d$, is equivalent to the field-theoretic conjecture for all fields of transcendence degree $d$ over $k$. It is known in dimension at most two, but remains open in dimensions at least three.