The Type (1) face orbit conjecture for K3-fibered threefolds

Let $Y$ be a smooth projective $3$-fold and let $f:Y\to\mathbb{P}^{1}$ be a K3 fibration such that $-K_Y=f^*\mathcal{O}(1)$. Let $\operatorname{Mov}^{e}(Y)$ denote the effective movable cone, and let Type (1) faces be the faces associated with the blowup of a smooth curve of genus $g$.

Type (1) face orbit conjecture. The pseudoautomorphism group $\operatorname{PsAut}(Y)$ acts with finitely many orbits on the Type (1) faces of $\operatorname{Mov}^{e}(Y)$ for every genus $g\geq 0$.

This assertion concerns the finiteness of birational models arising from Type (1) extremal contractions. The supplied text gives no resolution status for this statement, so it is recorded as open.