Anticanonical minimal models and Zariski decomposition

Birkar and Hu showed that if a pair $(X,\mathrm{\Delta })$ is lc and $K_X+\mathrm{\Delta }$ admits a birational Zariski decomposition, then $(X,\mathrm{\Delta })$ has a minimal model. Analogously, we prove that if a pair $(X,\mathrm{\Delta })$ is pklt and $-(K_X+\mathrm{\Delta })$ admits a birational Zariski decomposition, then $(X,\mathrm{\Delta })$ has an anticanonical minimal