Erdős–Fishburn triangular-lattice conjecture for few-distance configurations

Let $L_{\Delta}=\left\{\left(a+\frac{1}{2}b,\frac{\sqrt{3}}{2}b\right):a,b\in\mathbb{Z}\right\}$ be the triangular lattice. A configuration is $k$-optimal if it has the maximum possible number of points among configurations determining at most $k$ distinct distances. Erdős–Fishburn's conjecture. There exists at least one $k$-optimal configuration in $L_{\Delta}$ for all $k\geq 3$, and all $k$-optimal configurations are represented by subsets of $L_{\Delta}$ for all $k\geq 7$.