The projective-plane completion conjecture for dense Sidon sets

Let $G$ be an abelian group of order $n$, and let $S\subset G$ be a dense Sidon set, meaning that $|S|\geq (1-o(1))n^{1/2}$. Let $\mathcal{P}$ be a projective plane, and let $G$ act on it.

Projective-plane completion conjecture. The action is faithful, $|\mathcal{P}|=(1+o(1))n$, and for some point $p$ and line $\ell$,

$$S\subset\{g\in G:p^g\in\ell\}.$$

This is the main conjecture of the paper and is equivalent to the assertion that the development of $S$ can be completed to a projective plane by adding $o(|G|)$ points and lines. It is presented as open; the authors say they have no idea how to approach it and that it may be false. This candidate is merged with the prose restatement earlier in the paper.