The spectral reconstruction conjecture for symmetric matrices

Let $A$ be a real $n \times n$ symmetric matrix. For each $i$, write $A_i$ for the matrix obtained by deleting the $i$th row and column, and let $\operatorname{eigen}(A)$ denote the spectrum of $A$. Spectral reconstruction conjecture. There exists a subgroup $G(A) \subseteq O(n)$ such that a real symmetric matrix $B$ satisfies

$$\operatorname{eigen}(B)=\operatorname{eigen}(A)$$

and

$$\operatorname{eigen}(B_i)=\operatorname{eigen}(A_i)$$

for each $i$ if and only if $B=UAU^t$ for some $U \in G(A)$. The conjecture concerns whether the spectrum of a symmetric matrix together with the spectra of all principal vertex-deleted submatrices determines the matrix up to the prescribed orthogonal symmetries. The source notes that it is clearly true when $\operatorname{rank}(A)=1$, with $G(A)$ choosable as the diagonal sign group $\mathbb Z_2^n$; in other cases the group may be a subgroup of the permutation group $S_n$.