The Topological Period-Index Conjecture for finite CW complexes

Let $X$ be a topological space homotopy equivalent to a finite CW complex of dimension $2d$, and let $\alpha\in\operatorname{Br}_{top}(X)$.

Topological Period-Index Conjecture. One has

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

This is the naive topological analogue of the period-index conjecture for central simple algebras. The hypothesis is known to fail for general finite CW complexes when $d\geq 3$, although it holds for compact $\operatorname{spin}^c$ manifolds of dimension $6$; it is intended for restricted classes such as closed orientable manifolds, $\operatorname{spin}^c$ manifolds, and complex projective manifolds.