Hexagonal construction conjecture for maximum distance sets

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 equiangular hexagon has six sides in the lattice directions; its side lengths alternate between $k$ and $k+1$, while a regular hexagon has all six side lengths equal. Taking all triangular-lattice points in one of these hexagons produces an $m$-distance set for the number $m$ of distances that it determines. Hexagonal construction conjecture. Taking all points in an equiangular hexagon with side length alternating $k$ and $k+1$, or in a regular hexagon, will always give a maximum $m$-distance set. The paper reports confirmed examples for $m=3$ and $m=5$, but leaves the general assertion unresolved.