For what matroids $M$ does its set of circuits $\mathscr{C}_M$ satisfy $\small\forall C_1\in\mathscr{C}_M\exists C_2\in\mathscr{C}_M\setminus\{C_1\}:C_1\cap C_2\neq\emptyset$?

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 $\small \{C_1,C_2\}\in E(\mathscr{C}_M)\iff C_1\cap C_2\neq\emptyset$)