Unirationality conjecture for moduli components of smooth polarized Calabi–Yau pairs

Let $Q$ be a connected component of the coarse moduli space of triples $(X,E=E_1+\dots+E_n,\Theta)$ such that $X$ is a connected smooth projective complex variety, $E\in\lvert-K_X\rvert$ is a normal crossing divisor with a $0$-stratum and every $E_i$ is smooth, and $\Theta\subset X$ is an ample divisor not containing any $0$-stratum of $E$. Let $\overline{Q}$ denote the closure of $Q$ in the moduli space of stable pairs. Unirationality conjecture. There is a complete toric variety $T$ with a finite surjective map

$$T\to\overline{Q}.$$

This more precise formulation implies that every such component $Q$ is unirational. It concerns the expected toric structure underlying moduli of smooth polarized Calabi–Yau pairs; the parser supplies no evidence resolving the conjecture.