Special MMP for log canonical generalised pairs

We show that minimal models of $\mathbb{Q}$-factorial NQC log canonical generalised pairs exist, assuming the existence of minimal models of smooth varieties. More generally, we prove that on a $\mathbb{Q}$-factorial NQC log canonical generalised pair $ (X,B+M) $ we can run an MMP with scaling of an ample divisor which terminates, assuming that it admits an NQC weak Zariski decomposition or that $K_X+B+M$ is not pseudoeffective. As a consequence, we establish several existence results for minimal models and Mori fibre spaces.Comment: v4: Theorem 1.2 is new, proved in the appendix written jointly with Xiaowei Jiang; several other proofs slightly improve