Definability conjecture for lattice path matroids

A lattice path matroid is the matroid represented by a lattice path in the standard lattice-path construction. Let ${\mathit{MS}_{0}}$ denote monadic second-order logic over matroids without quantification over edge sets. Definability conjecture. The class of lattice path matroids can be characterised by a sentence in ${\mathit{MS}_{0}}$. Lattice path matroids have infinitely many excluded minors, but the paper proves that the class is pigeonhole; the conjecture would, together with the stated companion conjecture, imply decidability of its ${\mathit{MS}_{0}}$ theory with bounded branch-width.