John, you wrote

> $$\Phi(x,y) = I \otimes \mathcal{X}(x,x) \le \mathcal{X}(x,x) \otimes \Phi(x,y)$$

in your proof of the last Lemma. Did you actually mean

$$\Phi(x,y) = I \otimes \Phi(x,y) \le \mathcal{X}(x,x) \otimes \Phi(x,y)$$

or is there something I'm missing?