The finite abelian group bound for phylogenetic complexity

Let $G$ be a finite abelian group and let $n$ be a positive integer. The quantity $\phi(G,n)$ denotes the maximum degree needed to generate the corresponding toric ideal of phylogenetic invariants.

Finite abelian group bound. For any finite abelian group $G$ and any positive integer $n$,

$$\phi(G,n) \leq |G|.$$

The conjecture is motivated by the theorem that $I_{\mathbb{Z}_2,n}$ is generated by quadrics and by computational results. If true, it would imply that the phylogenetic complexity of every finite abelian group is bounded by its order.