The geometric cone conjecture in relative dimension two

Let $X\to S$ be a fibration of relative dimension at most two and let $(X,\mathrm{\Delta })$ be a klt pair for which $K_X+\mathrm{\Delta }{\equiv }_{S}0$. We show that there are only finitely many Mori chambers and Mori faces in the movable effective cone ${\mathcal{M}}^e(X\mathrm{/}S)$ up to the action of relative pseudo-automorphisms of $X\mathrm{/}S$ preserving $\mathrm{\Delta }$.