Gallardo–Grekos–Pihko conjecture on three-fold restricted sumsets

Let $n$ be an odd positive integer, let $\mathbb{Z}_n$ be the cyclic group of order $n$, and let $A\subseteq\mathbb{Z}_n$. Write $3^\wedge A$ for the set of sums of three distinct elements of $A$:

$$3^\wedge A=\{a_1+a_2+a_3:a_i\in A,\ a_i\ne a_j\text{ for }i\ne j\}.$$

Gallardo–Grekos–Pihko conjecture. There is a constant $c$ such that, whenever

$$|A|>\frac{2}{5}n+c,$$

one has

$$3^\wedge A=\mathbb{Z}_n.$$

Earlier results establish the conclusion for larger density thresholds, including $|A|>n/2$ for all integers $n\geq12$ except $n=15$, while the stated $2/5$ threshold remains the conjectural target.