Cheltsov–Shramov conjecture on Klein-group Mori fibre spaces over the projective line

For each integer $n\geq 0$, let $\mathcal{X}_n$ be the three-dimensional $\operatorname{PSL}_2(\mathbb{F}_7)$-Mori fibre space over $\mathbb{P}^1$ constructed above. The other variety under consideration is the product $\mathbb{P}^2\times\mathbb{P}^1$, also viewed as a three-dimensional $\operatorname{PSL}_2(\mathbb{F}_7)$-Mori fibre space over $\mathbb{P}^1$.

Cheltsov–Shramov conjecture. The varieties

$$\mathbb{P}^2\times\mathbb{P}^1 \quad\text{and}\quad \mathcal{X}_n,\qquad n\geq 0,$$

are the only $\operatorname{PSL}_2(\mathbb{F}_7)$-Mori fibre spaces over $\mathbb{P}^1$ in dimension $3$.

This is a classification conjecture for the equivariant Mori fibre spaces appearing in the construction. The source gives no resolution of it.