The eventual constancy recurrence for phylogenetic complexity

Let $G$ be a finite abelian group, and let $\phi(G,n)$ denote its phylogenetic complexity at level $n$.

Eventual constancy recurrence.

$$\phi(G,n+1)=\max(2,\phi(G,n)).$$

The paper describes this as a strengthening of its theorem that $\phi(G,\cdot)$ is eventually constant. It also notes that, apart from the case $G=\mathbb{Z}_2$ and $n=3$, the assertion is equivalent to constancy of $\phi(G,\cdot)$; it is presented as an open conjecture.