Relative McKernan–Prokhorov boundedness conjecture for log Calabi–Yau fibrations

Let $d$ and $r$ be positive integers, and let $\epsilon$ be a positive real number. A base-polarized fibration is a fibration $X\to(Z,A)$ with a polarization $A$ on the base. A $(d,r,\epsilon)$-log Calabi–Yau rationally connected fibration $(X,B+{\bf M})\to(Z,A)$ is the type of fibration defined in the source, with $X$ of dimension $d$, base polarization satisfying the $r$-boundedness condition, and the indicated $\epsilon$-log Calabi–Yau and rational connectedness properties.

Relative McKernan–Prokhorov conjecture. The set of base-polarized fibrations $X\to(Z,A)$ such that there exists a $(d,r,\epsilon)$-log Calabi–Yau rationally connected fibration $(X,B+{\bf M})\to(Z,A)$ is bounded.

This is presented as a relative generalization of the McKernan–Prokhorov conjecture and of boundedness results for rationally connected fibrations. The source gives no resolution evidence, so the conjecture is recorded as open.