For what matroids $M$ does there exist a monotone map $f:\{C:C\text{ is a circuit of }M\}\to\mathbb{N}$ such that for any circuits $C_1$ and $C_2$ of $M$ we have $f(C_1)<f(C_2)\implies C_1\cap C_2\neq\emptyset$?

Or equivalently for what matroids $M$ is the intersection graph of the circuits in $M$ connected?

Suppose we call matroids with this property "overlapping" then if two matroids $M_1$ and $M_2$ are overlapping and some circuit in $M_1$ is not disjoint to some circuit in $M_2$ then $M_1\oplus M_2$ is also overlapping, thus as a corollary every overlapping matroid must be a connected matroid.