Uniqueness conjecture for Brauer–Severi varieties of conic-bundle orders

Let $X$ be the Brauer–Severi variety of an order of type $F^2_3$, $F^2_4$, or $F^{2-}_5$, as described in the table in the source. Let $Y$ be another variety birational to $X$ over $\mathbb{P}^2$ and having the same anticanonical degree. Uniqueness conjecture. Then $Y\simeq X$. This is a geometric form of the conjectured uniqueness of the moduli object for the indicated types; the source gives no resolution status.