Classification conjecture for irreducible λ-quiddities on the subgroup generated by the golden ratio

Let $\varphi=\frac{1+\sqrt{5}}{2}$ be the golden ratio, and let $\langle\varphi\rangle=\{k\varphi:k\in\mathbb{Z}\}$ be the subgroup of $\mathbb{C}$ generated by $\varphi$. An irreducible $\lambda$-quiddity on $\langle\varphi\rangle$ is an irreducible sequence satisfying the $\lambda$-quiddity relation described in the paper. Classification conjecture. The set of irreducible $\lambda$-quiddities on $\langle\varphi\rangle$ is

$$\{(\varphi,\varphi,\varphi,\varphi,\varphi),(-\varphi,-\varphi,-\varphi,-\varphi,-\varphi),(0,k\varphi,0,-k\varphi),(k\varphi,0,-k\varphi,0); k\in\mathbb{Z}\}.$$

The paper notes that the two constant sequences of length five are examples, while a classification of irreducible solutions on this subgroup is not known; the conjecture proposes that these examples together with the displayed four-term families are all such solutions.