Quadratic-growth tau-sequence conjecture for octagon posets

Let $P$ be an octagon poset, and let $G(P)$ be its associated digraph. A strong $\tau$-sequence is a sequence of Laurent polynomials realizing the $R$-system through the relevant substitution; its degree growth is measured in the initial variables.

Octagon-poset tau-sequence conjecture. For each octagon poset $P$, the $R$-system associated with $G(P)$ admits a strong $\tau$-sequence consisting of irreducible Laurent polynomials whose degrees grow at most quadratically.

The conjecture is intended to imply singularity confinement and zero algebraic entropy for the $R$-systems associated with octagon posets. The supplied text gives no proof or status beyond presenting it as a conjecture.