Unimodular graded Poisson Hopf algebras

Let $A$ be a Poisson Hopf algebra over an algebraically closed field of characteristic zero. If $A$ is finitely generated and connected graded as an algebra and its Poisson bracket is homogeneous of degree $d \geq 0$, then $A$ is unimodular; that is, the modular derivation of $A$ is zero. This is a Poisson analogue of a recent result concerning Hopf algebras which are connected graded as algebras.