The twisted-product refinement of the spectral reconstruction conjecture

Let $S_n$ be the permutation group on $n$ letters and let $\mathbb Z_2^n$ denote the group of diagonal sign matrices. Twisted-product refinement. The group $G(A)$ in the spectral reconstruction conjecture can be chosen to be a twisted product of a subgroup of $S_n$ with $\mathbb Z_2^n$. This refinement specifies a proposed structure for the orthogonal symmetry group governing matrices with the same spectrum as $A$ and the same spectra for all principal vertex-deleted submatrices. The source gives no proof or resolution of this refinement; it follows the preceding conjectural reconstruction framework and supplements the stated rank-one and permutation-group examples.