Updated decomposition conjecture for near-regular matroids

A near-regular matroid is a matroid representable over every field except possibly $\operatorname{GF}(2)$. A signed-graphic matroid is the matroid associated with a signed graph. If $M_1$ and $M_2$ share a restriction isomorphic to $M(K_k)$, with $M_1$ graphic, their graph $k$-clique sum is the matroid obtained by taking the corresponding generalized parallel connection and deleting the common restriction; its dual is defined by dualizing this operation.

Updated decomposition conjecture. Every near-regular matroid can be obtained from signed-graphic matroids, the duals of signed-graphic matroids, or members of a finite set ${\cal C}$ by applying $1$-, $2$-, and $3$-sums, together with graph $k$-clique sums and their duals for $k\leq 4$.

This updates the earlier decomposition conjecture after the paper's obstruction to a direct analogue of Seymour's theorem. The authors note that general structural results would give bounded-rank perturbations under very large connectivity bounds, while this conjecture seeks a more refined specialized decomposition.