Log boundedness conjecture for K-moduli compactifications

Let $\mathcal{G}$ be a log bounded set of couples that is complete under dominant small modifications, and let $\widehat{\mathcal{G}}$ be its K-moduli completion: a couple belongs to $\widehat{\mathcal{G}}$ if it occurs as the special fiber of a family whose general fibers lie in $\mathcal{G}$ and whose fibers, for some coefficient vector $(c_1,\ldots,c_k)\in P$, are K-semistable weak log Fano pairs. Log boundedness conjecture. The set $\widehat{\mathcal{G}}$ is log bounded. The conjecture asserts that K-moduli completion preserves log boundedness, a finiteness property expected to be useful in the study of K-moduli compactifications. The source provides no resolution of the conjecture.