  In this work we generalize the entanglement of purification and its
conjectured holographic dual to conditional and multipartite versions of the
same, where the optimization defining the entanglement of purification is now
optimized in either a constrained way or over multiple parties. We separately
derive new constraints on both the conditional entanglement of purification and
its conjectured holographic dual object that match, further reinforcing the
likelihood of this conjecture. We also show that the multipartite objects we
define, despite obeying several of the same inequalities, are not holographic
duals of each other. Further, we find inequalities that are true only for the
bulk objects, and thus could provide additional consistency checks for states
dual to (semi)-classical bulk geometries.
