Classification of monohedral plane tilings with non-convex hexagons

Question

Is a classification of monohedral plane tilings with non-convex hexagons known or unknown? I prepare a seminar for talented basic school pupils and I would like to talk about monohedral tiling problem. Before that, I want to be sure what is known and what is not known. With triangles and quads it is clear, with convex pentagons it seems to be clear due to the proof from 2017, with convex hexagons it is also clear, with non-convex pentagons it is surely unclear due to the unsolved Einstein problem. With non-convex hexagons I was unable to find out if it is open or not, therefore I ask here. (If there is something known about non-convex pentagon tiling, I am also interested.).