Boundedness of B-canonical representations

Let $S$ be an $l$-dimensional variety with only canonical singularities and $K_S\sim 0$. Write

$$\rho_1:{\operatorname{Bir}}(S,0)\to {\operatorname{GL}}(H^0(S,K_S))$$

for the representation induced by birational automorphisms preserving the canonical divisor. For a finite-order automorphism $g\in {\operatorname{Aut}}(S,0)={\operatorname{Aut}}(S)$, let the order of $\rho_1(g)$ mean its order in ${\operatorname{GL}}(H^0(S,K_S))$.

Boundedness of $B$-canonical representations. There exist positive integers $B_l$ and $B'_l$ such that

$$|\rho_1({\operatorname{Bir}}(S,0))|\leq B_l$$

for every such $l$-dimensional variety $S$, and the order of $\rho_1(g)$ is at most $B'_l$ for every finite-order $g\in {\operatorname{Aut}}(S,0)={\operatorname{Aut}}(S)$.

These conjectures assert uniform boundedness of the images of birational automorphism groups and of the canonical representations of finite-order automorphisms in each dimension. The source introduces them as conjectural inputs for its main results; no resolution status is given here.