We know that $X$ and $Y$ are conditionally independent given $Z$ iff $\mathbb{P}(X,Y|Z) = \mathbb{P}(X|Z) \mathbb{P}(Y|Z)$.
The following condition is weaker than conditional independence: \begin{equation} \mathbb{E}_Z(\mathbb{P}(X|Z) \mathbb{P}(Y|Z)) = \mathbb{E}_Z(\mathbb{P}(X,Y|Z)), \end{equation}
in the sense that if $X$ and $Y$ are conditionally independent given $Z$, then $\mathbb{E}_Z(\mathbb{P}(X|Z) \mathbb{P}(Y|Z)) = \mathbb{E}_Z(\mathbb{P}(X,Y|Z))$, but the converse is apparently not necessarily true.
I am trying to find an example where this reverse direction does not hold. I.e., what is an example of a case when we have $\mathbb{E}_Z(\mathbb{P}(X|Z) \mathbb{P}(Y|Z)) = \mathbb{E}_Z(\mathbb{P}(X,Y|Z))$ but $X$ and $Y$ are conditionally dependent given $Z$?