Binary cocircuit conjecture for minimally vertically k-connected matroids

Let $k$ be an integer with $k\geq 5$. A matroid $M$ is minimally vertically $k$-connected if it is vertically $k$-connected and $M\setminus e$ is not vertically $k$-connected for every element $e$ of $M$.

Binary cocircuit conjecture. For each integer $k\geq 5$, every minimally vertically $k$-connected binary matroid with at least $2k+2$ elements has a $k$-cocircuit.

The paper proves the $k=4$ binary case and conjectures that the same phenomenon extends to all $k\geq 5$, owing to restrictions on circuit–cocircuit intersections in binary matroids. The general assertion remains open.