Occhetta–Wiśniewski's toric image conjecture

Let $X$ be a smooth projective variety, let $Z$ be a complete toric variety, and let

$$\phi:Z\to X$$

be a surjective map. Occhetta–Wiśniewski's conjecture. Then $X$ is a toric variety. The paper proves special cases for weak Fano threefolds of Picard rank $2$, extending earlier results for Fano threefolds; the general statement remains open in the supplied context.