Gabow's sequential symmetric exchange conjecture for matroid bases

Let $M$ be a matroid of rank $r$, and let $A$ and $B$ be bases of $M$. An ordering of a basis is a listing of its elements, so write

$$A=(a_1,\dots,a_r),\qquad B=(b_1,\dots,b_r).$$

Gabow's conjecture. There are orderings $A=(a_1,\dots,a_r)$ and $B=(b_1,\dots,b_r)$ such that

$$\{a_1,\dots,a_i,b_{i+1},\dots,b_r\}$$

and

$$\{b_1,\dots,b_i,a_{i+1},\dots,a_r\}$$

are bases for every $i=0,\dots,r$. This is the sequential symmetric exchange property, equivalently expressible through a cyclic ordering in which both bases form intervals and every $r$ cyclically consecutive elements form a basis. The conjecture concerns whether the extreme case of Gabow's symmetric-exchange decomposition can always be achieved.