Phylogenetic Invariants for $${mathbb{Z}_3}$$ Scheme-Theoretically

We study phylogenetic invariants of general group-based models of evolution with group of symmetries ({mathbb{Z}_3}). We prove that complex projective schemes corresponding to the ideal I of phylogenetic invariants of such a model and to its subideal ({I'}) generated by elements of degree at most 3 are the same. This is motivated by a conjecture of Sturmfels and Sullivant [14, Conj. 29], which would imply that ({I = I'}).