Nonrationality conjecture for the varieties $X_n$

Let $X_n$ be the varieties in the series of $G_K$-Mori fiber spaces over $\mathbb{P}^1$ whose general fibers are isomorphic to the degree-$2$ del Pezzo surface $S_K$. Nonrationality conjecture. The varieties $X_n$ are non-rational for $n\geq 2$. This predicts that the members of the constructed family from index $2$ onward are not rational, extending the paper's focus on proving nonrationality for del Pezzo fibrations by reduction modulo $p$. The supplied text gives no evidence that this conjecture has been resolved.