Classification of smooth Mori fibrations birational to the double cone variety

Let $U$ be the Fano variety of index $2$ and degree $5$ described in the surrounding text, and let $V_l\to U$ be the contraction associated with a curve $l$ on $U$. A smooth Mori fibration is a Mori fibration whose total space is smooth; two varieties are in the same class of birational equivalency when they are birational.

Classification conjecture. Any smooth Mori fibration birational to $U$ is biregular to either $U$ or $V_l$ for some curve $l$.

The conjecture is motivated by the failure of the maximal singularities method to handle all cases, including when $l$ has a double point. The supplied text gives no resolution status.