The following question looks similar to the double cover conjecture for graphs and regular matroids, but I am not sure if it has been studied for general matroids:
Is there a universal constant $c$ such that for every bridgeless matroid $M=(E,\mathcal{I})$ there is a multiset of circuits $C_1,C_2,\dots, C_k$ such that each element appears in at least one of them and in at most $c$?
For example, if the double cover conjecture for graphs is true then, the cycle matroid of any bridgeless graph satisfy the previous statement with $c=2$.
