Independence ratio conjecture for the distance graphs $\{1,k,k+7\}$

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 $k\geq8$ with $k\notin\{9,11,16,18,25\}$. The independence ratio conjecture.

$$\overline{\alpha}(\{1,k,k+7\})= \begin{cases} \frac{4k+27}{9k+72} & \text{if } k\equiv0\pmod9,\\ \frac49 & \text{if } k\equiv1\pmod9,\\ \frac{4k+28}{9k+72} & \text{if } k\equiv2\pmod9,\\ \frac{8k+21}{18k+63} & \text{if } k\equiv3\pmod9,\\ \frac{4k+29}{9k+72} & \text{if } k\equiv4\pmod9,\\ \frac{4k-2}{9k+9} & \text{if } k\equiv5\pmod9,\\ \frac{4k+30}{9k+72} & \text{if } k\equiv6\pmod9,\\ \frac{4k+26}{9k+72} & \text{if } k\equiv7\pmod9,\\ \frac{4k+31}{9k+72} & \text{if } k\equiv8\pmod9. \end{cases}$$

These values are conjectured from computed independence ratios for similar distance graphs; the source gives no proof or resolution for this family.