Birational equivalence conjecture for products of Severi–Brauer varieties

Let $P_i$ and $Q_j$ be Severi–Brauer varieties over a field $k$, and let

$$X=\prod_{i\in I}P_i,\qquad Y=\prod_{j\in J}Q_j$$

be their ordinary products. Write $\langle P_i:i\in I\rangle$ and $\langle Q_j:j\in J\rangle$ for the subgroups of $\operatorname{Br}(k)$ generated by their classes.

Product Severi–Brauer conjecture. The following are equivalent:

1. $\langle P_i:i\in I\rangle=\langle Q_j:j\in J\rangle$.
2. $X\times\mathbb{P}^{\dim Y}$ is birational to $Y\times\mathbb{P}^{\dim X}$.

If $\dim X=\dim Y$, these conditions are also equivalent to $X$ being birational to $Y$.

This generalizes Amitsur's conjecture from individual Severi–Brauer varieties to ordinary products. The statement is suggested by the birational product theorem proved earlier in the paper, but the supplied text does not state that the generalization has been resolved.