Uniqueness of the generic liftability matroid

Prove that every realisable matroid has a unique generic liftability matroid, namely that the row matroids of liftability matrices associated with generic realisations of the matroid’s truncation are independent of the irreducible component containing the generic realisation.

Background

For a matroid with a realisable truncation, the paper defines a generic liftability matroid as the row matroid of the liftability matrix associated with a generic point in an irreducible component of the truncation’s realisation space. Different irreducible components may, a priori, produce different generic liftability matroids.

The paper establishes that the generic liftability matroid is unique when the truncation’s realisation space is irreducible, but does not prove uniqueness in general. The conjecture would also imply that generic realisations across components yield the same liftability behavior in the elevation lattice.

References

It is currently open whether any matroid has more than one generic liftability matroid. Every realisable matroid has a unique generic liftability matroid.

— Matroid liftability and Oxley--Wang derived matroids, with connections to scene analysis  (2610.01854 - Dewar et al., 1 Oct 2026) in Section 3, subsection “Liftability matrices and matroids,” Conjecture 1 (labelled conj:genlift)

Similar to generic liftability matroids (see \Cref{conj:genlift}), the authors would conjecture that there exists a unique generic Oxley--Wang derived matroid for any realisable matroid. Every realisable matroid has a unique generic Oxley--Wang derived matroid.

— Matroid liftability and Oxley--Wang derived matroids, with connections to scene analysis  (2610.01854 - Dewar et al., 1 Oct 2026) in Section 5, subsection “Generic Oxley–Wang matroids and liftability,” Conjecture 2 (labelled conj:gom)