Sturmfels–Sullivant degree-bound conjecture for the 3-Kimura model

Let $T$ be a tree, and let $X(T,G)$ denote the variety associated to the group-based model for a finite group $G$. For $G=\mathbb{Z}_2\times\mathbb{Z}_2$, this is the 3-Kimura model, whose defining ideal consists of the phylogenetic invariants vanishing on the model.

Sturmfels–Sullivant's 3-Kimura conjecture. The ideal of the 3-Kimura model is generated in degree at most $4$ for any tree $T$.

This conjecture arose from numerical evidence for small trees and predicts a uniform degree bound for generators of the defining ideals. The statement is supported by the theorem that the projective scheme associated to the 3-Kimura model can be defined by an ideal generated in degree at most $4$, but the conjecture as stated for the ideal of the model remains open in the supplied source.