The Paley complementation-set exclusivity conjecture

Let $G$ be the Paley graph of prime order $p$, namely $G=\operatorname{Cay}(\mathbb{Z}_p,S)$ where $S$ is the set of quadratic residues modulo $p$. Let $a,b\in\mathbb{Z}_p^*$ have orders $4$ and $k$, respectively, with $a-1$ a quadratic residue, and define

$$S_1=a\langle b\rangle\cup\langle b\rangle,\qquad S_2=-a\langle b\rangle\cup\langle b\rangle.$$

A set is a complementation set of $G$ in the sense of the paper. Paley complementation-set conjecture. At most one of $S_1$ and $S_2$ is a complementation set of $G$. Moreover, if $p\equiv5\pmod {16}$, $S_1$ is not a complementation set, and if $p\equiv13\pmod {16}$, $S_2$ is not a complementation set. The conjecture is explicitly presented among statements suggested by computer testing; no resolution is supplied.