Monotone boundedness from bounded disjoint finite suprema

Determine whether a demicontinuous normed lattice in which every disjoint subset of the positive cone whose finite suprema lie in the unit ball is order bounded must be monotonically bounded.

Background

The paper characterizes monotone completeness and monotone boundedness using directed sets, dominability, and disjoint subsets. It notes that the proposed implication has a positive answer under the sufficiently many projections (SMP) assumption. The question asks whether the SMP hypothesis can be omitted when demicontinuity and the stated order-boundedness condition on disjoint subsets are assumed.

References

Assume that $F$ is demicontinuous and such that every disjoint $D\subset F_{+}$ such that $D{\vee}\subsetB_{F}$ is order bounded in $F$. Is $F$ monotonically bounded?

— Variants of order semicontinuity in Banach lattices  (2609.03070 - Bilokopytov, 2 Sep 2026) in Question immediately following Theorem 2.?? (the theorem labeled \ref{awt}), Section 5, “Monotone boundedness and completeness”