Rainbow circuit conjecture for regular matroids

Let $M$ be a simple matroid of rank $n-1$, and let $c$ be a colouring of its ground set $E(M)$ with $n$ colours. A circuit is rainbow when no two of its elements have the same colour. Rainbow circuit conjecture for regular matroids. If $M$ is regular and every colour class has size at least $2$, then $M$ contains a rainbow circuit of size at most

$$\left\lceil \frac{n}{2} \right\rceil.$$

The analogous conjecture for arbitrary simple rank-$(n-1)$ matroids is refuted in the paper by uniform matroids. The proposed regular-matroid statement is motivated by the fact that the result holds for graphic and cographic matroids, but it remains open.