  Let $Y$ be a smooth projective $3$-fold admitting a K3 fibration $f : Y
\rightarrow \mathbb{P}^1$ with $-K_Y = f^*\mathcal{O}(1)$. We show that the
pseudoautomorphism group of $Y$ acts with finitely many orbits on the
codimension one faces of the movable cone if $H^3(Y,\mathbb{C})=0$, confirming
a special case of the Kawamata--Morrison--Totaro cone conjecture. In [CCGK16],
[P18], and [CP18], the authors construct log Calabi-Yau 3-folds with K3
fibrations satisfying the hypotheses of our theorem as the mirrors of Fano
3-folds.
