Nonconfinement conjecture for the coefficient-free R-system of the five-element poset

Let $P$ be the poset shown in Figure~, and let $G(P)$ be its associated graph. Consider the coefficient-free $R$-system associated with $G(P)$. A strong $\tau$-sequence and the singularity confinement property are the corresponding structural and dynamical properties of this $R$-system. Nonconfinement conjecture. The coefficient-free $R$-system associated with $G(P)$ does not possess a strong $\tau$-sequence and does not have the singularity confinement property. The conjecture is motivated by computer calculations through fourteen iterations, during which no irreducible factor disappeared; those calculations do not establish nonconfinement for all future iterations.