Necessity of monotonicity for local finiteness of the glued sum

Determine whether the monotonicity condition (MC8) is required to prove that the sum of a modular connected system is locally finite.

Background

The paper proves that the sum lattice of a modular connected system is locally finite under axiom (MC8), the monotonicity condition. The proof uses (MC8) to establish a strict inequality in Lemma 4.4, which is then used to show that the intervals of skeleton elements associated with a lattice element are finite in Lemma 4.40.

The unresolved issue is whether local finiteness of the glued sum follows without this axiom, rather than whether the current proof can simply be shortened. The question concerns the necessity of the structural hypothesis itself.