Conjecture on the maximum 7- and 8-distance sets in triangular lattices

Let a triangular lattice be the set of points with coordinates $a(1,0)+b(\frac12,\frac{\sqrt3}{2})$, where $a,b\in\mathbb Z$. An $m$-distance set is a set of lattice points whose pairwise distances take at most $m$ values, and a maximum $m$-distance set has largest possible size. A regular hexagon with side length $2$ is understood as the corresponding hexagonal region of the triangular lattice. Maximum 7- and 8-distance conjecture. The maximum $7$-distance and $8$-distance sets have $16$ and $19$ points, respectively. The $7$-distance set can only be realized by taking all triangular-lattice points in a regular hexagon with side length $2$ and removing three mutually adjacent or three mutually non-adjacent corners. The $8$-distance set can only be realized by taking all points in that regular hexagon. These are presented as conjectures based on the authors' constructions; no proof or resolution is supplied.