Gabow's cyclic basis conjecture

Let $\mathcal{M}$ be a matroid and let

$$B_1=\{b_1,\ldots,b_r\},\qquad B_2=\{b'_1,\ldots,b'_r\}$$

be two bases. Gabow's conjecture. There exist permutations $\sigma$ and $\sigma'$ such that in the sequence

$$(b_{\sigma(1)},\ldots,b_{\sigma(r)},b'_{\sigma'(1)},\ldots,b'_{\sigma'(r)}),$$

every set of $r$ cyclically consecutive elements forms a basis. The conjecture strengthens the two-base case of White's basis sequence exchange problem by requiring a cyclic ordering with all consecutive $r$-element windows independent as bases.