For pseudoeffective projective pairs, termination of one flip sequence implies termination of all, conditional on a conjecture about Nakayama-Zariski decomposition in the MMP.
Moraga, Termination of pseudo-effective 4-fold flips , arXiv:1802.10202 0= 2018
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Nakayama-Zariski decomposition and the termination of flips
For pseudoeffective projective pairs, termination of one flip sequence implies termination of all, conditional on a conjecture about Nakayama-Zariski decomposition in the MMP.