Farber's basis pair graph connectivity conjecture

Let $M$ be a matroid with ground set $E(M)$ and rank $r(M)$. Its basis pair graph $G(M)$ has as vertices the ordered triples $(A_1,A_2,A_3)$, where $A_1$ and $A_2$ are disjoint bases of $M$ and $A_3=E(M)-(A_1\cup A_2)$. Two vertices are adjacent when they differ by switching a pair of elements in two different sets.

Farber's conjecture. The basis pair graph of any matroid is connected.

This conjecture was posed by M. Farber, who proved it for transversal matroids; the source also notes results for graphic and cographic matroids. It remains open in general.