Mori structures conjecture for double spaces of index 2

Let $U$ be a double space of index 2. A Mori structure means one of the Mori fibrations birationally associated with $U$ described in the preceding construction, including $U$ itself and fibrations $V_l/\mathbb{P}^1$ obtained from curves $l$.

Mori structures conjecture. The Mori structures of $U$ are precisely $U$ itself and

$$V_l/\mathbb{P}^1$$

for all curves $l$ of degree $2$ and genus $1$.

This conjecture classifies the Mori structures arising on a double space of index $2$; the supplied text does not state whether it has been proved or disproved.