The missing axiom conjecture for matroid representability in MSOL

Let a finite matroid be a finite set $E^{\mathcal{M}}$ equipped with a rank function

$$r^{\mathcal{M}}\colon \mathcal{P}(E^{\mathcal{M}})\to\mathbb{Z}^{+}\cup\{0\}.$$

Let MSOL denote monadic second-order logic for matroids, whose structures use finite ground sets, subsets, cardinalities, and rank functions. Missing axiom conjecture. There is no finite set of sentences, $\mathcal{K}$, in MSOL such that $(E^{\mathcal{M}},r^{\mathcal{M}})$ is a representable matroid if and only if it satisfies the rank axioms and every sentence in $\mathcal{K}$. This formalizes the claim that representability cannot be finitely axiomatized in MSOL. The paper proves finite axiomatizability for representability over fixed finite fields conditional on Rota's conjecture, while proving non-finite-axiomatizability over any fixed infinite field within the relevant framework; the general representability conjecture remains open.