The boundedness of B-representations conjecture

Let $d$ and $I$ be positive integers, and let $(X,B)$ be a projective $d$-dimensional klt log Calabi–Yau pair such that $I(K_X+B)\sim 0$. The birational automorphism group $\operatorname{Bir}(X,B)$ acts on the one-dimensional space $H^0(I(K_X+B))$. Boundedness of B-representations conjecture. There is a constant $b\coloneqq b(d,I)$ such that, for every such pair, the image of

$$\operatorname{Bir}(X,B)\longrightarrow \operatorname{GL}(H^0(I(K_X+B)))\simeq \mathbb{K}^*$$

is finite and has order at most $b$. This conjecture predicts uniform boundedness of the birational automorphism actions on pluricanonical sections; it is used in the paper for analogous higher-coregularity results, while its general validity remains open.