Coalgebra-symmetry conjecture for Liouville integrability of Poisson maps

Let $T\colon n\mapsto n+1$ be a Poisson map admitting a coalgebra symmetry $(\mathfrak{A},\Delta)$. Let the generators of the coalgebra symmetry satisfy the closure relation referred to in the source. Coalgebra-symmetry conjecture. The map $T$ is Liouville integrable if and only if the evolution of those generators is Poisson--Liouville integrable. This conjecture proposes that the integrability properties of an underlying Poisson map are completely governed by those of the associated dynamical system on the generators of its coalgebra symmetry; the source presents it as a conjectural generalization of the examples discussed, and no resolution is supplied.