Classification conjecture for Klein simple group Mori fiber spaces over the projective line

Let $G_K\cong \mathrm{PSL}_2(\mathbb{F}_7)$ be the Klein simple group, and let $X_n/\mathbb{P}^1$ be the $G_K$-Mori fiber spaces described above. A $G_K$-Mori fiber space is a three-dimensional Mori fiber space over $\mathbb{P}^1$ equipped with a compatible $G_K$-action. Classification conjecture. The fibrations $\mathbb{P}^1\times\mathbb{P}^2/\mathbb{P}^1$ and $X_n/\mathbb{P}^1$, for $n\geq 0$, are the only $G_K$-Mori fiber spaces over $\mathbb{P}^1$ in dimension $3$. This conjecture proposes a complete classification of the three-dimensional $G_K$-Mori fiber spaces over $\mathbb{P}^1$; its resolution would clarify the possible embeddings of the Klein simple group into the rank-three Cremona group. The supplied text gives no evidence that it has been resolved.