The Shokurov-polytope version of the relative cone conjecture

Let $f:(X,\Delta)\to S$ be a klt Calabi–Yau fiber space. Let $\operatorname{Eff}(X/S)$, $\operatorname{Mov}(X/S)$, and $\operatorname{Amp}(X/S)$ denote the effective, movable, and ample cones, respectively. Shokurov-polytope cone conjecture. There exists a polyhedral cone $P_M\subset \operatorname{Eff}(X/S)$ such that

$$\bigcup_{g\in\operatorname{PsAut}(X/S,\Delta)}g\cdot P_M\supset \operatorname{Mov}(X/S),$$

and there exists a polyhedral cone $P_A\subset \operatorname{Eff}(X/S)$ such that

$$\bigcup_{g\in\operatorname{Aut}(X/S,\Delta)}g\cdot P_A\supset \operatorname{Amp}(X/S).$$

The conjecture is proposed as a more tractable formulation related to Shokurov polytopes; its general validity remains open.