Enveloping-matroid realization for even symmetric matroids

Ascertain whether for every even symmetric matroid M on E = [n] ∪ [n]* there exists a matroid N on E such that the bases of M coincide with the bases of N restricted to transversals and almost-transversals, i.e., B(M) = B(N) ∩ (T ∪ A), where T is the set of transversals and A is the set of almost-transversals.

Background

The paper raises an inverse problem for antisymmetric matroids: given an antisymmetric matroid M on E, is there a matroid N on E whose bases restrict to B(M) on transversals and almost-transversals? This is confirmed when M is representable over a field.

The authors note that the analogous question for even symmetric matroids remains unresolved. Establishing such an enveloping matroid for even symmetric matroids would parallel known correspondences between matroids and antisymmetric matroids and connect to the literature on enveloping matroids.