General congruence transformation to a diagonal-noise adjacency representation

Let $\boldsymbol{\boldsymbol{\text{?}}}$ denote the covariance and adjacency objects in the source construction, with observed and unobserved index sets $J_O$ and $J_U$. Let $W'$ be the adjacency matrix in the root-confounder data-generation model, and let $Q$ satisfy the displayed matrix equation and the requirements referenced in the source.

General diagonal-noise representation conjecture. There exists a general adjacency matrix

$$W”=QW'Q^{-1}$$

of the displayed block form, with $[W]_{J_U,J_O} ot ot\to 0$ and $[W]_{J_U} ot ot\to 0$, such that $W”$ with independent idiosyncratic variables represents the same observed data, preserving $Sigma_{11}= Sigma_Y= E(YY^T)$; equivalently, the corresponding noise covariance $Omega”= E( epsilon” epsilon”^T)$ is diagonal, where $epsilon”=Q epsilon'$.

The statement is presented as future work after a special-case construction, and the source supplies no evidence of resolution. It concerns whether latent-confounding representations can be transformed into observationally equivalent representations with diagonal idiosyncratic noise.