For what matroids $M$ can its circuits be indexed $C_1,\ldots C_n$ s.t. $\forall k<n(C_k\cap C_{k+1}\neq\emptyset$)?

Or equivalently for what matroids $M$ is the intersection graph of $\mathscr{C}_M$ connected?

(this is the graph $\small G$ with $\small V(G)=\mathscr{C}_M$ and $\small \{C_1,C_2\}\in E(\mathscr{C}_M)\iff C_1\cap C_2\neq\emptyset$)