The period-index conjecture for Brauer groups of complex varieties

Let $X$ be a smooth, connected, projective variety over $\mathbf C$. For a Brauer class $\alpha in \operatorname{Br}(X)$, its period and index are denoted by $\operatorname{per}(\alpha)$ and $\operatorname{ind}(\alpha)$, respectively.

Period-index conjecture. For every $\alpha in \operatorname{Br}(X)$,

$$\operatorname{ind}(\alpha) \divides \operatorname{per}(\alpha)^{\dim X-1}.$$

This is the projective geometric form of the period-index conjecture for Brauer groups of function fields. The supplied paper’s abstract states that the complex-analytic version is false for infinitely many Brauer classes on a general complex torus of dimension at least three, while the status of the precise projective statement here is not specified in the supplied text.