Limit-point spectrum conjecture for maximal cliques in d-Paley graphs

Fix an integer $d\ge 2$. For each odd prime power q\equiv1{\ifmmode\text{\rm\ (mod~d)}\else\discretionary{}{}{\text{ }}\rm(mod~d)\fi}, let $r_q$ be the cardinality of a maximal clique in the $d$-Paley graph $GP(q^d,d)$, and consider the sequences $\{r_q/q\}$. Write $\operatorname{rad}(n)$ for the product of the distinct prime divisors of $n$. Limit-point spectrum conjecture. The set of limit points of all such sequences is

$$\{0\}\cup\left\{\frac{1}{m}:\operatorname{rad}(m)\mid\operatorname{rad}(d)\right\}.$$

Theorem~ establishes that every listed value occurs as a limit point; the conjectural part is that no other limit points occur. This is intended as a converse to the construction of large maximal cliques.