Amitsur's birationality conjecture for Severi–Brauer varieties

Let $F$ be a field, and let $A$ and $B$ be central simple algebras over $F$. Write $SB(A)$ and $SB(B)$ for their associated Severi–Brauer varieties, and let $[A]$ and $[B]$ denote their Brauer classes in the Brauer group $\operatorname{Br}(F)$. Amitsur's conjecture.

$$SB(A)\sim SB(B)\quad\Longleftrightarrow\quad \langle[A]\rangle=\langle[B]\rangle\text{ in }\operatorname{Br}(F),$$

that is, the two Brauer classes generate the same cyclic subgroup of $\operatorname{Br}(F)$. The conjecture characterizes birational equivalence of Severi–Brauer varieties in terms of the cyclic subgroups generated by their Brauer classes. It remains open for central simple algebras of prime power index.