Large Circuits in Binary Matroids of Large Cogirth, II

LetF7denote the Fano matroid andMbe a simple connected binary matroid such that every cocircuit ofMhas size at leastd⩾3. We show that ifMdoes not have anF7-minor,M≠F*7, andd∉{5, 6, 7, 8}, thenMhas a circuit of size at least min{r(M)+1, 2d}. We conjecture that the latter result holds for alld⩾3.