Khashin's Mori-structure conjecture for the double space U

Let $U$ be the double space considered in the source, and let ${\mathcal S}$ denote the family of curves used there to construct the fibrations $V_l/\mathbb{P}^1$. Khashin's conjecture. The variety $U$ has exactly the following Mori structures: $U$ itself and $V_l/\mathbb{P}^1$ for every $l\in\mathcal S$, giving a two-dimensional family of different del Pezzo fibrations. In particular, $U$ is not rational because it has no conic bundles.

The conjecture concerns the classification of Mori structures on $U$ and would imply its non-rationality. The source says that Khashin's earlier attempted proof of non-rationality contains mistakes and that no reliable proof of non-rationality was known there.