The period-index conjecture for $C_d$ fields

Let $K$ be a $C_d$ field, meaning that every homogeneous form of degree $e$ in more than $e^d$ variables over $K$ has a nontrivial zero, and let $\alpha\in\operatorname{Br}(K)$. The period-index conjecture for $C_d$ fields.

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

This is the folk conjecture described as being supported by the period-index results known at the time. It predicts a uniform bound on the index in terms of the period for Brauer classes over $C_d$ fields.