Eventual linearity of ultradiscrete Painlevé solutions

Let $Y_m$ and $Z_m$ be solutions of ultradiscrete Painlevé VI without parity variables, with fixed parity variables $\frak{y}_m=\frak{z}_m=-1$ for all $m$, and let $Q,A_i,B_i$ be the parameters subject to the constraint $B_3+B_4+A_1+A_2=Q+A_3+A_4+B_1+B_2$. Consider the linear forms

$$Y_m=\alpha'm+\beta',\qquad Z_m=\alpha'm+\gamma'$$

for $m\leq m_0'$ and

$$Y_m=(Q-\alpha)m+\beta,\qquad Z_m=\alpha m+\gamma$$

for $m\geq m_0$, where $m_0,m_0'\in\mathbb{Z}$ and $\alpha,\beta,\gamma,\alpha',\beta',\gamma'\in\mathbb{R}$. Eventual linearity conjecture. Every solution $Y_m,Z_m$ for $m\in\mathbb{Z}$ has this form for some $m_0,m_0'$ and parameters satisfying the conditions in equations

$$and$$

. The conjecture is supported by several numerical solutions and asserts that every solution is linear in both tails, with potentially different linear expressions as $m\to-\infty$ and $m\to+\infty$.