Shokurov's toric-model conjecture for maximal log Calabi–Yau threefolds

Let $(X,\Delta_X)$ be a maximal log Calabi–Yau pair, meaning a pair with the maximality property considered in the paper, and suppose that $X$ is a rational $3$-fold. A toric model is a birational model of the pair that is toric in the relevant sense.

Shokurov's toric-model conjecture. Every maximal log Calabi–Yau pair $(X,\Delta_X)$ whose underlying variety $X$ is a rational $3$-fold has a toric model.

This conjecture gives a criterion for characterising maximal log Calabi–Yau pairs with a toric model and originates in work of Shokurov. The paper's main theorem proves a special case. The stated hypotheses are essential: the claim fails without rationality, in dimensions other than three, or when the condition $K_X+\Delta_X\sim 0$ is weakened to $K_X+\Delta_X\sim_{\mathbb Q}0$.