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.
After revision:  Finally, we prove that the 3-Fall Coloring problem is NP-complete on bipartite graphs with diameter at most four, and prove that NP-completeness for diameter three would also imply NP-completeness of 3-Precoloring Extension on bipartite graphs with diameter 3.