The period-index conjecture over fields of finite transcendence degree

Let $k$ be an algebraically closed field, a $C_1$ field, or a $p$-adic field, and set $e=0$, $1$, or $2$, respectively. Let $K$ be a field of transcendence degree $n$ over $k$. For $\alpha\in\operatorname{Br}(K)$, write $\operatorname{per}(\alpha)$ for its period and $\operatorname{ind}(\alpha)$ for its index.

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

$$\operatorname{ind}(\alpha)\mid\operatorname{per}(\alpha)^{n-1+e}.$$

This conjecture unifies the known period-index results for function fields over algebraically closed, finite, and local fields. The surrounding discussion records several cases and partial bounds, while the stated general bound remains open.