Fusion hierarchies, T-systems and Y-systems for the dilute A2(2) loop models on a strip

We study the dilute $A_2^{(2)}$ loop models on geometry of a strip width $N$. Two families boundary conditions are known to satisfy Yang-Baxter equation. Fixing condition two ends leads four models. construct fusion hierarchy commuting transfer matrices for model as well its $T$- and $Y$-systems, these with generic crossing parameter $\lambda$. For $\lambda/\pi$ rational thus $q=-e^{4i\lambda}$ root unity, we prove linear relation satisfied by fused that closes into finite system. The relations allow us compute leading terms in large-$N$ expansion free energy, namely bulk energies. These found be agreement numerical data obtained small present work complements previous (A. Morin-Duchesne, P.A. Pearce, J. Stat. Mech. (2019)) investigated periodic conditions.