The period-index conjecture for unramified Brauer classes

Let $X$ be a smooth projective variety over an algebraically closed field $k$. For $b1 \in \operatorname{Br}(X)$, let $\mathrm{per}(b1)$ and $\mathrm{ind}(b1)$ denote the period and index of its restriction to the generic point.

Unramified period-index conjecture. There exists a positive integer $e$ such that, for all $b1 \in \operatorname{Br}(X)$,

$$\mathrm{ind}(b1) \text{ divides } \mathrm{per}(b1)^e.$$

This is the unramified form of the period-index problem for Brauer classes on smooth projective varieties. The corresponding global bound is known in transcendence degree two, but the higher-dimensional case remains open; the source notes that even a uniform power bound is not known for a single field of transcendence degree at least three.