Suppose $(Y,B_Y)$ is a sub-klt pair which is a birational model of a klt pair $(X,B)$: there exists a birational morphism $\pi:(Y,B_Y) \rightarrow (X,B)$ such that $\pi^{*}(K_X+B)=K_Y+B_Y$. We know that such pairs satisfy the basepoint free theorem (See Theorem 2.2 in "Basepoint free theorems, saturation, b-divisors and canonical bundle formula" by Fujino). So it's natural to wonder if such sub-klt pairs satisfy the other fundamental theorems of the MMP, namely the cone and contraction theorems(EDIT: note that the naive choice $cont_C= \pi \circ cont_{\pi(C)}$ is not the right one), the rationality theorem etc. Does anybody know if that is the case?