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