White's basis-sequence equivalence conjecture

Let $(B_1,\dots,B_k)$ and $(B'_1,\dots,B'_k)$ be sequences of $k$ bases of a matroid. Two sequences are compatible if the union of the $B_i$ as a multiset equals the union of the $B'_i$ as a multiset, and they are equivalent if one can be obtained from the other by a composition of symmetric exchanges. White's conjecture. Two sequences of $k$ bases are equivalent if and only if they are compatible. The compatibility condition is an evident necessary condition; the source records the assertion as open in general.