Laurent irreducibility conjecture for toric-digraph R-systems

Let $\Lambda\subset\mathbb{Z}^2$ be a rank-two lattice containing none of $e_1=(1,0)$, $e_2=(0,1)$, or $e_2\pm e_1$, and let $G(\Lambda)$ be the resulting toric digraph with vertex set $V=\mathbb{Z}^2/\Lambda$. Define $Y_v(t)$ recursively by the toric-digraph recurrence in the paper, with initial variables $\mathbf{x}_V=(x_v)_{v\in V}$.

Toric-digraph Laurent irreducibility conjecture. The rational functions $Y_v(t)$ are pairwise coprime irreducible polynomials in $\mathbf{x}_V$; more precisely, they are irreducible and coprime as Laurent polynomials, up to monomial factors.

This is the Laurent phenomenon suggested by the authors' computations. They additionally conjecture that these Laurent polynomials are actual polynomials, but the supplied statement asserts pairwise coprimality and irreducibility in the Laurent setting.