Erdős–Fishburn conjecture on maximum distance sets in triangular lattices

Let an $m$-distance set be a collection of points in the Euclidean plane such that the distances between any two points take at most $m$ possible values. A triangular lattice is the set of points with coordinates $a(1,0)+b(\frac12,\frac{\sqrt3}{2})$, where $a,b\in\mathbb Z$. A maximum $m$-distance set is an $m$-distance set with the largest possible cardinality. Erdős–Fishburn conjecture. For any $m\geq 3$, at least one maximum $m$-distance set lies within triangular lattices. If $m\geq 7$, then maximum $m$-distance sets can only be found in triangular lattices. This conjecture motivates the study of distance sets in triangular lattices; the source gives no resolution of either assertion.