A question from Jim Geelen about matroid circuits

Guest post by Jim Geelen $\newcommand{\del}{\,\backslash\,}$ $\newcommand{\con}{/}$ $\newcommand{\bF}{\mathbb{F}}$ $\newcommand{\cL}{\mathcal{L}}$ $\newcommand{\lt}{<}$ $\newcommand{\gt}{>}$ $\DeclareMathOperator{\ex}{ex}$ $\DeclareMathOperator{\rank}{rank}$ $\DeclareMathOperator{\PG}{PG}$ $\DeclareMathOperator{\AG}{AG}$ $\DeclareMathOperator{\BB}{BB}$

I was asked a question recently, by Anna Lubiw and Vinayak Pathak, which led me to the following conjecture. It looks quite natural, so I wonder whether it is already known.

Conjecture: In any rank-$r$ matroid $M$, that has a circuit, there is a circuit $C$ such that each component of $M\con C$ has rank at most $r/2$.

I would be happy with $r/2$ replaced by $\alpha r$ for any $\alpha \lt 1$. The conjecture holds for graphic and cographic matroids.

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