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Colourings and flows are well-known dual notions in Graph Theory. In turn,the definition of flows in graphs naturally extends to flows in orientedmatroids. So, the colour-flow duality gives a generalization of Hadwiger'sconjecture about graph colourings, to a conjecture about coflows of orientedmatroids. The first non-trivial case of Hadwiger's conjecture for orientedmatroids reads as follows. If $\mathcal{O}$ is an $M(K_4)$-minor free orientedmatroid, then $\mathcal{O}$ has a now-where $3$-coflow, i.e., it is$3$-colourable in the sense of Hochst\"attler-Ne\v{s}et\v{r}il. The class ofgeneralized series parallel ($GSP$) oriented matroids is a class of$3$-colourable oriented matroids with no $M(K_4)$-minor. So far, the onlytechnique towards proving that all orientations of a class $\mathcal{C}$ of$M(K_4)$-minor free matroids are $GSP$ (and thus $3$-colourable), has been toshow that every matroid in $\mathcal{C}$ has a positive coline. Towards provingHadwiger's conjecture for the class of gammoids, Goddyn, Hochst\"attler, andNeudauer conjectured that every gammoid has a positive coline. In this work wedisprove this conjecture by exhibiting an infinite class of strict gammoidsthat do not have positive colines. We conclude by proposing a simpler techniquefor showing that certain oriented matroids are $GSP$. In particular, we recoverthat oriented lattice path matroids are $GSP$, and we show that orientedcobicircular matroids are $GSP$.

Oriented cobicircular matroids are $GSP$