Is intersection of $k \ge 3$ graphic matroids in P?
I think it is still NP-complete, by a reduction from Hamiltonian paths in bipartite graphs with two degree-one vertices and all other vertices having degree three. (This is just the same as finding Hamiltonian cycles through a specified edge in a cubic bipartite graph — replace the specified edge by two leaves.)

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.