The maximality conjecture for circle constructions

Let $R$ have only prime factors $p$ satisfying $p\equiv 1\pmod 4$. Consider the two plane integral point sets supplied by the circle constructions: $\operatorname{circle}(R)$, with $2\tau(R)$ points on a circle of radius $R$ together with its center, and $\widetilde{\operatorname{circle}}(R)$, with $2\tau(R)$ points on a circle of radius $R/2$. Here $\tau(R)$ denotes the number of divisors of $R$. Circle-construction maximality conjecture. Both of these plane integral point sets are maximal. The claim concerns maximality of the two explicitly constructed families; the supplied text gives no proof or resolution.