4 problems

Recognition conjecture. The class of near-regular matroids is polynomial-time recognizable.

Recognition conjecture. The class ${\cal C}$ of near-regular signed-graphic matroids is polynomial-time recognizable.

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…

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…