De Loera–Lee–Margulies–Miller conjecture on minor-minimal non-weakly-orientable matroids

Let $M\!I_n$ be the infinite family of matroids constructed in the paper, indexed by integers $n\geq 0$. A matroid is weakly orientable when its Bland–Jensen linear system has a solution over $F_2$; it is minor-minimal non-weakly-orientable when it is non-weakly orientable but every proper minor is weakly orientable. The authors conjecture that

Minor-minimality conjecture. For all integers $n\geq 0$, $M\!I_n$ are minor-minimal with respect to being non-weakly-orientable. Hence, weak-orientability cannot be described by a finite list of excluded minors.

The family $M\!I_n$ is an infinite family of non-weakly-orientable matroids with growing rank and corank. The authors checked explicitly that $M\!I_n$ is minor-minimal for $n\leq 2$; the conjecture would imply the earlier Bland–Jensen conjecture that weak orientability cannot be characterized by a finite list of excluded minors.