Tait's perfect-difference-set extension conjecture

Let $p$ be a sufficiently large prime, and let $G=C_{p^2+p+1}$. Suppose $S\subset G$ is a Sidon set of size $p$. A perfect difference set of size $p+1$ is a subset $S'\subset G$ of size $p+1$ whose nonzero differences represent every nonzero element of $G$ exactly once.

Tait's perfect-difference-set extension conjecture. The set $S$ is contained in some perfect difference set $S'$ of size $p+1$.

This is presented as an additional open problem, attributed to Michael Tait by personal communication. It asks whether every sufficiently large near-perfect Sidon set in the indicated cyclic group extends to a perfect difference set.