Reflections for non-Type-1-or-3-to-6 monotiles

Determine whether every convex pentagonal monotile without line symmetry in the set T_{only - 2} ∪ (T_{7to13} ∩ T_1^C) can generate a tiling without using reflected tiles.

Background

The paper identifies convex pentagonal monotiles belonging only to Type 2, or to Types 7–13 while not belonging to Type 1, as cases for which the representative tilings require reflected tiles. Types 14 and 15 are excluded because their fixed shapes evidently cannot tile without reflections. For the remaining non-line-symmetric cases, the paper does not know whether reflection-free tilings exist.