Weighted exchange distance of basis pairs

Two pairs of disjoint bases P1=(R1,B1) and P2=(R2,B2) of a matroid M are called equivalent if P1 can be transformed into P2 by a series of symmetric exchanges. In 1980, White conjectured that such a sequence always exists whenever R1∪B1=R2∪B2. A strengthening of the conjecture was proposed by Hamidoune, stating that the minimum length of an exchange is at most the rank of the matroid. We propose a weighted variant of Hamidoune's conjecture, where the weight of an exchange depends on the weights of the exchanged elements. We prove the conjecture for several matroid classes: strongly base orderable matroids, split matroids, graphic matroids of wheels, and spikes.