Determine whether AT-tile and BH-tile admit tilings other than the Tile(1,1)-derived non-periodic tiling

Determine whether the tile set consisting of the two concave polygons AT-tile and BH-tile admits tilings other than the non-periodic tiling \(\mathscr{T}_s\).

Background

The pair AT-tile and BH-tile is derived through the correspondence with the S8/S9S_8/S_9 substitution and is known to generate the non-periodic tiling Ts\mathscr{T}_s. Whether this pair admits any additional tiling patterns is unresolved.

References

A tile set consisting of two types of concave polygons, AT-tile and BH-tile, is known to generate $\mathscr{T}_s$, but whether it admits other tilings remains unknown.

— Tile sets consisting of two types of concave polygons derived from periodic tilings corresponding to non-periodic tilings with hat and turtle tiles  (2609.09603 - Sugimoto, 9 Sep 2026) in Section 5.3, Properties of tile sets containing AH-tile, BH-tile, AT-tile, and BT-tile