The elementary subgroup conjecture for Sidon sets in $\mathbf{F}_p^2$

Let $p$ be prime and let $G=\mathbf{F}_p^2$. Suppose $S\subset G$ is a Sidon set of size $p$. Define

$$T=G\setminus(S-S)\cup\{0\}.$$

Elementary subgroup conjecture. The set $T$ is a subgroup.

If true, the results cited in the source would imply that $S$ is a parabola. The authors describe this as a basic case they are unable to solve, so it remains open.