4 problems

Obstruction excluded-minor conjecture. If $\Delta$ has an obstruction, then $M(\Delta)$ is an excluded minor for $\mathcal{BO}$.

Ingleton's critical-graph conjecture. If $\Delta$ is a critical graph with no obstructions, then $M(\Delta)$ is an excluded minor for $\mathcal{BO}$.

Ingleton's excluded-minor conjecture. Ingleton conjectured that an infinite set of matroids forms excluded minors for $\mathcal{BO}$.

Asymptotic base-orderability conjecture. Almost all matroids are base-orderable, in the sense that the proportion of matroids that are base-orderable tends to $1$ asymptotically.