Efficient pigeonhole conjecture for bicircular and gain-graphic matroids

Let $H$ be a finite group. Efficient pigeonhole conjecture. The class of bicircular matroids is efficiently pigeonhole, and the class of $H$-gain-graphic matroids is efficiently pigeonhole. The surrounding discussion presents these as conjectures about classes for which efficient pigeonhole structure is expected; no resolution is supplied in the paper excerpt.