Support-recovery incidence-variety conjecture for sparse phase retrieval

Let $N$ be a positive integer, and let $S,S'\subset[0,N-1]$ be non-equivalent subsets of size $K$, where equivalence means belonging to the same orbit under the dihedral group $D_{2N}$. Let $a_x$ be the periodic auto-correlation of $x$, let $L_S$ and $L_{S'}$ be the subspaces of vectors supported in $S$ and $S'$, respectively, and define the incidence variety

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

Support-recovery conjecture. If $|S-S|=|S'-S'|\geq K$, then $I_{S,S'}$ has dimension strictly less than $K$. This would imply that a generic vector supported on $S$ cannot share its auto-correlation with a vector supported on a non-equivalent set $S'$, so the support is generically recoverable up to dihedral equivalence. The conjecture is presented as the support-recovery step in the paper's computational verification strategy.