K-moduli conjecture for polarized Fano fibrations

Fix a positive integer $n$ and a real number $\epsilon>0$. A polarized Fano fibration has dimension $n$ and weighted volume at least $\epsilon$ when its dimension and weighted volume satisfy those bounds; K-polystability is the stability notion used in the source.

K-moduli conjecture for polarized Fano fibrations. Given a positive integer $n$ and $\epsilon>0$, there exists a projective moduli space parametrizing K-polystable polarized Fano fibrations of dimension $n$ and weighted volume at least $\epsilon$.

This predicts a projective compactified moduli theory for K-polystable polarized Fano fibrations, extending known moduli constructions for K-polystable Fano varieties and related metric objects. The conjecture is open.