# Is intersection of $k \ge 3$ graphic matroids in P?

It is known that intersection of three general matroids is NP-hard (source), which is done via reduction from Hamiltonian cycle. The reduction uses one graphic matroid and two connectivity matroids.

A special case of a problem I am working on can be solved by intersection multiple graphic matroids, but I haven't been able to find, whether this problem is in P.

Question: Is it known? Can someone please refer me to a paper or something?

(Note: I have asked this question on Computer Science and was referred here.)

To reduce from Hamiltonian paths to graphic matroid intersection, use one graphic matroid to force the subgraph you choose to be a spanning tree (true of every path) and two more graphic matroids, one on each side of the bipartition, to force the subgraph to have degree two at each degree-three vertex and to have an edge at each degree-one vertex. These are the graphic matroids of a graph with disjoint copies of $K_3$ for each degree-three vertex and $K_2$ for each degree-one vertex.