When does a matroid $M$ have a set of circuits $\mathcal{C}$ with a connected intersection graph i.e. when is the graph $G$ with$V(G)=\mathcal{C}$ and adjacencies $\{A,B\}\in E(G)\iff A\cap B\neq\emptyset$ connected?

This is equivalent to charactering the matroids with a partial ear decomposition i.e. the matroids with circuits that can be indexed $C_1,\ldots C_n$ so we get that $\forall 0<i<n\exists j<i:C_i\cap C_j\neq\emptyset$ (where note this indexing is not necessarily injective i.e. there might exist $i\neq j$ with $C_i=C_j$)

Suppose we call matroids with this property special now if two matroids $M_1$ and $M_2$ are special and some circuit in $M_1$ is not disjoint to some circuit in $M_2$ then $M_1\oplus M_2$ is also special, so as a corollary every special matroid must be a connected matroid. What do "special" matroids look like?