Generic uniqueness conjecture for sparse phase retrieval on finite abelian groups

Let $A$ be a finite abelian group, let $V$ be the vector space of functions $A\to\mathbb K$, and let $L_S$ be the subspace of functions supported in $S\subseteq A$. The periodic auto-correlation is

$$a_x[\ell]=\sum_{\ell'\in A}x[\ell']\overline{x[\ell+\ell']},$$

and $D_A$ is the intrinsic-symmetry group: $(S^1\times A)\ltimes\mathbb Z_2$ over $\mathbb C$, or $(\{\pm1\}\times A)\ltimes\mathbb Z_2$ over $\mathbb R$. Finite-abelian-group uniqueness conjecture. If

$$|S-S|>|S|$$

and $x\in L_S$ is generic, then $a_x=a_{x'}$ implies that $x'$ is obtained from $x$ by an action of $D_A$. This generalizes the sparse periodic phase-retrieval conjecture from cyclic groups to finite abelian groups; the source gives no resolution, so it remains open.