Gabow's symmetric exchange conjecture

Prove that, for every rank-r matroid with two disjoint bases A and B, the pair (A,B) can be transformed into (B,A) by exactly r symmetric exchanges.

Background

A symmetric exchange simultaneously replaces an element e of A\B with an element f of B\A in the first basis and replaces f with e in the second basis, with both resulting sets required to remain bases.

The paper identifies Gabow's conjecture as a long-standing unresolved problem concerning exchange sequences between pairs of bases in unoriented matroids. It is presented as a related problem motivating comparison with the oriented-matroid conjecture.