Independence ratio conjecture for the distance graphs $\{1,6,k\}$

Let $G(S)$ be the distance graph on the integers with generating set $S$, and let $\overline{\alpha}(S)$ denote its maximum density of an independent set. Let $k>6$ with $k\notin\{7,10,12,17\}$. The independence ratio conjecture.

$$\overline{\alpha}(\{1,6,k\})= \begin{cases} \frac{3k}{7k+7} & \text{if } k\equiv 0\pmod{7},\\ \frac{3}{7} & \text{if } k\equiv 1\pmod{7},\\ \frac{3k+1}{7k+7} & \text{if } k\equiv 2\pmod{7},\\ \frac{3k-2}{7k+7} & \text{if } k\equiv 3\pmod{7},\\ \frac{3k+2}{7k+7} & \text{if } k\equiv 4\pmod{7},\\ \frac{3k-1}{7k+7} & \text{if } k\equiv 5\pmod{7},\\ \frac{3}{7} & \text{if } k\equiv 6\pmod{7}. \end{cases}$$

The source notes that the proposed extremal sets are proved for all residue classes except $k\equiv3\pmod 7$, so the displayed formula remains conjectural in that class.