Consider an independence system $(E,\mathcal{I})$, and the corresponding polytope:
$P(E,\mathcal{I}):=\operatorname{conv.hull}\{ x^S ~|~S\in \mathcal{I}\}$
where $x^S \in \{0,1\}^E$ denotes the characteristic vector of $S$. A rank inequality has the form $\sum_{i \in S} x_i \le r(S)$, where $S\subseteq E$ and $r(S):=\max\{ |A| : A \subseteq S,~ A \in \mathcal{I} \}$. Let's say that the nonnegativity bounds $-x_i \le 0,~i \in E$ are trivial.
(1) When is it true that all nontrivial facets of $P(E,\mathcal{I})$ come from rank inequalities?
It is known, for example, that if $(E,\mathcal{I})$ is a matroid, or the intersection of two matroids, then all nontrivial facets of $P(E,\mathcal{I})$ come from rank inequalities.
(2) Does the converse hold, i.e., if all nontrivial facets of $P(E,\mathcal{I})$ come from rank inequalities, then $(E,\mathcal{I})$ can be written as the intersection of two matroids?