The permutation reconstruction conjecture for simple eigenvectors of hypomorphic matrices

Let $A$ and $B$ be two hypomorphic matrices, meaning that $B=\Sigma(A)$ for a hypomorphism $\Sigma$ as in the surrounding reconstruction setting. Let $\lambda_i$ be a simple eigenvalue of $A$, and write $\operatorname{eigen}_{\lambda_i}(A)$ and $\operatorname{eigen}_{\lambda_i}(B)$ for the corresponding one-dimensional eigenspaces. Permutation reconstruction conjecture. There exists a permutation matrix $\tau$ such that

$$\tau\,\operatorname{eigen}_{\lambda_i}(A)=\operatorname{eigen}_{\lambda_i}(B).$$

The claim proposes that every simple eigenspace is reconstructible up to a permutation of coordinates. The supplied context establishes related reconstruction results for simple eigenvalues whose eigenspaces are not orthogonal to the all-ones vector, but does not resolve this full assertion for arbitrary simple eigenvalues.