The Quadratic Zariski Problem for anisotropic quadrics

Let $X$ and $Y$ be anisotropic quadrics of the same dimension over a field $k$. A rational morphism means a rational map between these varieties, defined on a dense open subset. Quadratic Zariski Problem. If there exist rational morphisms

$$X \dashrightarrow Y \quad\text{and}\quad Y \dashrightarrow X,$$

then $X$ and $Y$ should be birational. This asks whether stable birationality forces birationality for smooth anisotropic quadrics; the problem is stated as open in the source, although the general Zariski problem is false without the quadratic hypotheses.