The phylogenetic complexity conjecture for the Klein four-group

Let $G=\mathbb{Z}_2\times\mathbb{Z}_2$ be the Klein four-group, and let its phylogenetic complexity be denoted by $\phi(G)$.

Klein four-group phylogenetic complexity conjecture. The phylogenetic complexity of $G$ is

$$\phi(G)=4.$$

The paper states that this follows from the preceding recurrence conjecture and presents it as an open consequence concerning a specific finite abelian group.