White's compatibility conjecture for symmetric exchanges

Prove that any two compatible ordered pairs of bases in a matroid can be connected by a sequence of symmetric exchanges.

Background

Two ordered pairs of bases (A,B) and (A',B') are called compatible when they have the same intersection and the same union, conditions that are necessary for transformation by symmetric exchanges.

White conjectured that these necessary compatibility conditions are also sufficient. The paper cites this conjecture as one of the long-standing exchange-sequence questions related to its oriented-matroid program.

References

The conditions $A\cap B=A'\cap B'$ and $A\cup B= A'\cup B'$, called {\it compatibility}, are clearly necessary for such a transformation, and White conjectured that they are also sufficient.

— A note on embracing exchange sequences in oriented matroids  (2511.14526 - Bérczi et al., 18 Nov 2025) in Section 3 (Matroid exchange analogues)