The Circuit-Cocircuit Intersection Conjecture for Intersection Size $\le 7$

A circuit-cocircuit intersection, or a CCI for short, of a matroid is the intersection of a circuit and a cocircuit. Oxley conjectured (1992) that a matroid with a CCI of size $k\ge 4$ has a CCI of size $k-2$. We show that the conjecture holds for $k\le 7$.