The Antieau-Williams topological period-index conjecture

Let $X$ be a finite CW complex of dimension $2d$, and let $\alpha\in\operatorname{Br}(X)$.

Antieau-Williams' conjecture.

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

This is the proposed topological analogue of the period-index conjecture for fields; the factor $2d$ reflects that a complex algebraic variety of dimension $d$ has an underlying space with a $2d$-dimensional cell decomposition. Antieau and Williams disproved this conjecture, so the asserted divisibility fails in general.