Generic signal-recovery conjecture with known support

Let $S\subset[0,N-1]$ and let $L_S$ be the subspace of signals supported in $S$. Let $D_S$ be the subgroup of intrinsic symmetries $D$ that preserves $L_S$. Generic signal-recovery conjecture. If $|S-S|>|S|$, $x i L_S$ is generic, and $x' i L_S$ satisfies $a_x=a_{x'}$, then

$$x'=g\cdot x$$

for some $g i D_S$. Thus, when the support is known, the Fourier magnitude determines a generic signal up to an intrinsic symmetry preserving that support. The source presents this as an independent component of the main conjecture and gives no proof, so it remains open.