Generic uniqueness conjecture for sparse periodic phase retrieval

Let $N$ be even, let $K$ be the sparsity, and let $x$ be a $K$-sparse generic signal whose periodic auto-correlation is $a_x$. An intrinsic symmetry is an element of the group $D$ of intrinsic symmetries, consisting of the symmetries that preserve the periodic auto-correlation. The generic uniqueness conjecture. If $K e 0$, $K e 1$, $K e N-1$, and $K e N$, and if $K\text{ or }N-K$ is at least $3$, then, under the stated support condition that $a_x$ has more than $K$ non-zero entries, $a_x=a_{x'}$ \implies that $x'$ is obtained from $x$ through an intrinsic symmetry. Equivalently, the periodic auto-correlation mapping is injective, up to intrinsic symmetries, for almost all signals. This is presented as the main conjecture of the paper; the source gives no resolution, so its status remains open.