Before revision:  for every k \geq 3. Since k-List Coloring was already known to be NP-complete on complete bipartite graphs, and polynomial for k=2 on general graphs, the only remaining open problems are List 3-Coloring and 3-Precoloring Extension  when d=3.
After revision:  for every k \geq 3. Since k-List Coloring was already known to be NP-complete on complete bipartite graphs, and polynomial for k=2 on general graphs, the only remaining open problems are List 3-Coloring and 3-Precoloring Extension problems when d=3.