  This article is part of an ongoing investigation of the two-dimensional
Jacobian conjecture. In the first paper of this series, we proved the
generalized Magnus' formula. In this paper, inspired by cluster algebras, we
introduce a sequence of new conjectures including the remainder vanishing
conjecture. This makes the generalized Magnus' formula become a useful tool to
show the two-dimensional Jacobian conjecture. In the forthcoming paper(s), we
plan to prove the remainder vanishing conjecture.
