Brumer–Rosen conjecture on primary components of Brauer groups

Let $F$ be a field and let $(\operatorname{Br}F)_p$ denote the $p$-primary component of its Brauer group. Brumer–Rosen conjecture. For each prime $p$, one of the following holds:

1. $(\operatorname{Br}F)_p=0$.
2. $(\operatorname{Br}F)_p$ contains a nonzero divisible subgroup.
3. $p=2$ and $(\operatorname{Br}F)_2$ is an elementary abelian $2$-group.

This conjecture would imply that finite nonzero Brauer groups are $2$-groups of the indicated elementary abelian type; the source gives no general resolution.