The CCI-envelope form of Oxley's circuit-cocircuit intersection conjecture

A circuit-cocircuit intersection (CCI) is a subset of a matroid's ground set that is the intersection of a circuit and a cocircuit. A CCI-envelope is a matroid that contains a CCI of size $k$ and whose ground set has size $2k-2$, for an integer $k\ge4$. Oxley's circuit-cocircuit intersection conjecture in CCI-envelope form. Every CCI-envelope of a size-$k$ CCI has a CCI of size $k-2$. This is an equivalent reformulation of Oxley's conjecture, since every CCI has a minor on $2k-2$ elements in which it is both a circuit and a cocircuit. The paper proves the conjecture for CCI size $k\le7$, while the general case remains open.