Characterization of closed upper sets in the MacNeille completion of the free monoid

Let $\Lambda$ be a well-founded conditional lattice, and let $Z$ be an upper set of the free monoid $\Lambda^{\ast}$. The MacNeille completion identifies closed upper sets with upper sets satisfying four specified rules.

Closed-upper-set characterization. An upper set $Z$ of $\Lambda^{\ast}$ is closed if and only if it satisfies the four rules.

This gives a syntactic characterization of the closed upper sets in the MacNeille completion of $\Lambda^{\ast}$. The supplied text does not state whether this result is conjectural or has been proved; the surrounding discussion presents related results as consequences of a theorem, so the status should be checked against the paper.