Amitsur's equal-degree conjecture for Brauer–Severi varieties

Let $X$ and $Y$ be Brauer–Severi varieties, and let $A$ and $B$ be the corresponding central simple $k$-algebras. Assume

$$\operatorname{deg}(A)=\operatorname{deg}(B).$$

Amitsur's conjecture. If $A$ and $B$ generate the same cyclic subgroup of $\mathrm{Br}(k)$, then $X$ and $Y$ are birational.

This is the converse to the birationality implication proved by Amitsur: birational Brauer–Severi varieties have corresponding algebras generating the same subgroup of the Brauer group. The source does not specify whether the conjecture has been resolved in general.