Maximal Matroids in Weak Order Posets

Let $cX$ be a family of subsets of a finite set $E$. A matroid on $E$ is called an $cX$-matroid if each set in $cX$ is a circuit. We consider the problem of determining when there exists a unique maximal $cX$-matroid in the weak order poset of all $cX$-matroids on $E$, and characterizing its rank function when it exists.