Matrix product formula for $U_q(A^{(1)}_2)$-zero range process

The $U_q(A^{(1)}_n)$-zero range processes introduced recently by Mangazeev, Maruyama and the authors are integrable discrete and continuous time Markov processes associated with the stochastic $R$ matrix derived from the well-known $U_q(A_n^{(1)})$ quantum $R$ matrix. By constructing a representation of the relevant Zamolodchikov-Faddeev algebra, we present, for $n=2$, a matrix product formula for the steady state probabilities in terms of $q$-boson operators.