Hamidoune's bounded symmetric exchange conjecture

Prove that any two compatible basis pairs in a rank-r matroid can be transformed into one another using at most r symmetric exchanges.

Background

Hamidoune's conjecture is an optimization version of the symmetric exchange problem that encompasses the settings considered by Gabow and White. It concerns ordered pairs of bases with equal intersection and union.

The conjecture asserts a universal rank-based upper bound on the number of symmetric exchanges needed to connect compatible pairs, and is presented as a long-standing conjecture related to the paper's main oriented-matroid question.