Embedding-Bias in Conditional Independence Testing

To test conditional independence of $X$ and $Y$ given a text or an image $Z$, one conditions on an embedding $ψ(Z)$ in place of $Z$. The embedded test is valid if $Z$ is independent of $X$ or of $Y$ given $ψ(Z)$, which cannot be confirmed from data, and when this fails, the rejection probability und