Generalized Amitsur conjecture for generalized Severi–Brauer varieties

Let $F$ be an infinite field, and let $A$ and $B$ be central simple algebras over $F$ of the same degree. For each $r<\deg(A)$, write $F_r(A)$ and $F_r(B)$ for the function fields of the corresponding generalized Severi–Brauer varieties. Generalized Amitsur conjecture. If $[A]$ and $[B]$ generate the same cyclic subgroup of $\operatorname{Br}(F)$, then

$$F_r(A)\cong F_r(B)$$

for every $r<\deg(A)$. This extends Amitsur's conjecture from ordinary Severi–Brauer varieties to generalized ones; the source presents it as a new conjecture and proves it in the cases described in the surrounding discussion, including when $B\cong A^{op}$, while it remains open in general.