Hilbert polynomial of the Kimura 3-parameter model

Buczy'{n}ska and Wi'{s}niewski showed that for the Jukes Cantor binary model of a 3-valent tree the Hilbert polynomial depends only on the number of leaves of the tree and not on its shape. We ask if this can be generalized to other group-based models. In this paper we consider the Kimura 3-parameter model and show that the generalization of the statement about the Hilbert polynomials to the Kimura 3-parameter model is not possible as the Hilbert polynomial depends on the shape of a 3-valent tree.