Non-existence of primitive formally dual subsets of size 8 in the group $\mathbb{Z}_8^2$

Let $G=\mathbb{Z}_8^2$ and let a subset $S\subset G$ be primitive if it is not contained in a proper coset of a subgroup of $G$ and generates $G$. A subset is formally dual if there is a pairing for which its character-sum identities satisfy the formal duality condition. Non-existence conjecture. There is no primitive formally dual subset $S\subset\mathbb{Z}_8^2$ with

$$|S|=8.$$

The graph search algorithm terminated without finding such a subset, but the authors present the computational non-existence result as a conjecture because they do not provide a readily checkable certificate; a more sophisticated proof remains to be found.