On page 12 of the paper Enumeration of chord diagrams on many intervals and their non-orientable analogs" by Alexeev, Andersen, Penner, and Zograf is a list of polynomials which are a refinement of the first few Motzkin polynomials listed in OEIS A055151.

Can anyone provide a proof that this relation between the two sets of polynomials holds for all higher degrees in general?