The detachable-pair classification conjecture for 3-connected matroids

Let $M$ be a $3$-connected matroid with $|E(M)| \ge 13$.

Detachable-pair classification conjecture. Either $M$ has a detachable pair; $M$ is a spike; $M \cong M(K_{3,k})$ or $M \cong M^*(K_{3,k})$ for some $k \ge 5$; $M$ is the cycle matroid of a wheel or a whirl; $M$ can be constructed by attaching wheels to a spike with tip and cotip; or $M$ or $M^*$ can be constructed by attaching wheels with common spokes $s$ and $t$, and then deleting $s$.

This conjecture classifies the exceptional $3$-connected matroids with no detachable pair once the ground set has at least thirteen elements. The preceding discussion identifies several families with no detachable pairs, including spikes, wheel and whirl matroids, and matroids obtained by attaching wheels; the classification remains unproved in the supplied text.