The independent set conjecture for triangle-free 3-connected matroids

Let $M$ be a triangle-free $3$-connected matroid with a simple $3$-connected minor $N$. An element of $M$ is $N$-contractible if contracting it yields a matroid with an $N$-minor. Then $M$ has an independent set $I$ of $N$-contractible elements such that

$$|I|=r(M)-r(N).$$

Independent set conjecture. If $M$ is a triangle-free $3$-connected matroid with a simple $3$-connected minor $N$, then $M$ has an independent set $I$ of $N$-contractible elements such that $|I|=r(M)-r(N)$. This would generalize the corresponding result for graphic matroids and provide a stronger structural statement for non-graphic matroids; its status is not established in the source.