Kawamata–Morrison cone conjecture for K-trivial fiber spaces

Let $f:X\to S$ be a $K$-trivial fiber space, meaning a normal $\mathbb Q$-factorial klt pair $(X,\Delta)$ endowed with a proper surjective morphism with connected fibres such that $K_X+\Delta\equiv_f0$. Write $\overline{\operatorname{Mov}}^e(X/S):=\overline{\operatorname{Mov}}(X/S)\cap\operatorname{Eff}(X/S)$ and $\operatorname{Nef}^e(X/S):=\operatorname{Nef}(X/S)\cap\operatorname{Eff}(X/S)$. Kawamata–Morrison cone conjecture. There exists a rational polyhedral cone $\Pi$ such that

$$\overline{\operatorname{Mov}}^e(X/S)=\bigcup_{g\in\operatorname{PsAut}(X/S)}g^*\Pi,$$

and

$$\Pi^\circ\cap(g^*\Pi)^\circ=\emptyset$$

for every $g\in\operatorname{PsAut}(X/S)$, except when $g^*=1$ in $\operatorname{GL}(N^1(X/S))$. There also exists a rational polyhedral cone $\Pi'$ satisfying the analogous fundamental-domain conditions for the action of $\operatorname{Aut}(X/S)$ on $\operatorname{Nef}^e(X/S)$. The conjecture predicts rational polyhedral fundamental domains for the movable and nef cones; in the source paper, the stated version is proved under the hypotheses described in the abstract, while the supplied span itself gives no resolution status for the general formulation.