Converse of the transversal-restriction map for general symmetric matroids

Determine whether every symmetric matroid N on the ground set E = [n] ∪ [n]* arises as the transversal restriction of some antisymmetric matroid; specifically, decide whether for every symmetric matroid N = (E, B_N) there exists an antisymmetric matroid M = (E, B_M) such that B_N = B_M ∩ T, where T is the set of transversals (size-n subsets of E containing no skew pair).

Background

Proposition 3.16 shows that taking the transversal bases of an antisymmetric matroid M on E produces a symmetric matroid. Theorem 3.17 proves that the converse holds for even symmetric matroids, providing a unique antisymmetric extension whose transversal bases recover the given even symmetric matroid.

However, whether this converse extends beyond the even case to all symmetric matroids is not established in the paper. Resolving this would clarify how broadly antisymmetric matroids encompass symmetric matroids via their transversal bases.