A conjecture of Welsh revisited

Welsh conjectured that for any simple regular connected matroid M, if each cocircuit has at least 12(r(M)+1) elements, then there is a circuit of size r(M)+1. This conjecture was proven by Hochstättler and Jackson in 1997. In this paper, we give a shorter proof of this conjecture based solely on matroid-theoretical arguments. Let M be a simple, connected, regular matroid and let C∈C(M), where |C|≤min{r(M),2d−1}. We show that if |C∗|≥d≥2,∀C∗∈C∗(M) where C∩C∗=0̸, then there is a circuit D such that D△C is a circuit where |D△C|>|C|.