The properness conjecture for K-moduli spaces of K-polystable Fano varieties

Fix a positive integer $n$ and a volume $V$, and let $X^{\rm Kps}_{n,V}$ be the good moduli space of $n$-dimensional K-polystable $\mathbb{Q}$-Fano varieties of volume $V$.

Properness conjecture. The good moduli space $X^{\rm Kps}_{n,V}$ is proper.

Properness is the main missing property in the algebraic construction of the K-moduli space. The source reformulates it valuatively as the existence, after finite base change, of K-semistable fillings for families over punctured curves.