Known-support incidence-variety conjecture for sparse phase retrieval

Let $N$ be a positive integer, let $S\subset[0,N-1]$, and let $D_S$ be the subgroup of the dihedral group $D_{2N}$ preserving $S$. Let $L_S$ be the subspace of vectors supported in $S$, and let $a_x$ denote periodic auto-correlation. Define the incidence variety

$$I_S=\{(x,x')\mid a_x=a_{x'}\}\subset L_S\times L_{S'}.$$

Known-support recovery conjecture. If $|S-S|>|S|$, then $I_S$ has dimension $|S|$ and degree $2|D_S|$. This is the signal-recovery step for generic signals whose support is known; the factor $2|D_S|$ records the global sign and the dihedral symmetries preserving the support. The source gives no resolution of this conjecture.