Nonfaithfulness conjecture for C^1 actions of finite-index mapping class groups

Let $S$ be a connected orientable surface with finitely many punctures, finitely many boundary components, and genus at least $2$, and let $G$ be a finite-index subgroup of the mapping class group of $S$. Nonfaithfulness conjecture. Any $C^1$ action of $G$ on the circle cannot be faithful.

This conjecture concerns the smoothability and faithfulness of circle actions of finite-index subgroups of mapping class groups. The paper explains that, if all finite-index subgroups of the relevant mapping class groups have trivial first cohomology, its theorem on direct-product actions would imply the conjecture for surfaces of genus at least $4$; the general conjecture is presented as open.