Before revision:  Finally, we prove that the 3-Fall Coloring problem is NP-complete on bipartite graphs with diameter at most 4, and give a polynomial reduction from 3-Fall Coloring on bipartite graphs with diameter 3 to 3-Precoloring Extension on bipartite graphs with diameter 3. The latter result implies that if 3-Fall Coloring is NP-complete on these graphs, then the complexity gaps mentioned above for List k-Coloring and k-Precoloring Extension would be closed .
After revision:  Finally, we prove that the 3-Fall Coloring problem is NP-complete on bipartite graphs with diameter at most 4, and give a polynomial reduction from 3-Fall Coloring on bipartite graphs with diameter 3 to 3-Precoloring Extension on diameter three, thus closing the previously mentioned open cases .