Whittle's 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 represented by a signed graph; $1$-, $2$-, and $3$-sums are the standard matroid sum operations.

Whittle's decomposition conjecture. Every near-regular matroid can be obtained from signed-graphic matroids, their duals, and members of some finite set by applying $1$-, $2$-, and $3$-sums.

This is proposed as an analogue of Seymour's decomposition theorem for regular matroids. The paper presents it as an open conjecture and shows that a satisfactory decomposition theory for near-regular matroids faces an obstruction.