Sturmfels–Sullivant phylogenetic complexity conjecture

Let $G$ be a finite abelian group, and let $K_{n,1}$ be a claw tree with $n$ leaves. Write $\varphi(G,n)=d$ for the least natural number such that the ideal associated with $K_{n,1}$ for the general group-based model with group $G$ is generated in degree $d$, and define the phylogenetic complexity by

$$\varphi(G)=\sup_nvarphi(G,n).$$

Sturmfels–Sullivant's conjecture. For every abelian group $G$,

$$\varphi(G)\leq |G|.$$

This conjecture predicts a uniform degree bound for generators of the ideals of claw-tree models, independent of the number of leaves. The paper states that it was suggested by numerical results and remains open there; a weaker version is used as support for the authors' method.