The subgroup-union conjecture for dense maximal Sidon sets

Let $G$ be a group of order $n$, and let $S\subset G$ be a dense maximal Sidon set, meaning that $S$ is maximal among Sidon subsets and has size at least $(1-o(1))n^{1/2}$. Define

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

Subgroup-union conjecture. The set $T$ is the union of $O(1)$ subgroups.

This is stated as a weaker companion to the projective-plane conjecture. In one cited example, $T$ is the union of three subgroups, and the authors suggest this may be the worst case. They explicitly state that these conjectures are open and may be false.