The harder restricted-tile conjecture for hexagonal mosaics

For a knot, let a hexagonal $r$-mosaic be a knot diagram assembled on the hexagonal $r$-mosaic arrangement, and let the interior tiles be the tiles away from the boundary. The four three-crossing tiles are tiles 23, 24, 25, and 26. Harder conjecture. Every knot can be created as a hexagonal $r$-mosaic for some $r$, while restricting the interior tiles to only these four three-crossing tiles. This is presented as a stronger open variant of the easier restricted-tile conjecture, removing the two crossing-free tile types.