Skeletons of modular locally finite lattices

Determine whether every locally finite lattice with finite covers is the skeleton of some modular, locally finite lattice with finite covers.

Background

The paper studies modular connected systems consisting of a skeleton lattice, an overlap tolerance, finite modular complemented blocks, and connecting maps. The skeleton is required to be locally finite with finite covers, but it is not itself required to be modular. The author notes that every finite lattice is known to occur as the dissection skeleton of some finite modular lattice, leaving unresolved whether the analogous assertion holds for all locally finite lattices with finite covers.

A positive answer would characterize the possible skeleton lattices arising from modular, locally finite lattices with finite covers and would extend the finite result cited from Herrmann’s work to the broader infinite setting considered in the paper.