\] Let $(G,P)$ be a quasi-lattice ordered group and $X$ a $\tilde\phi$-injective compactly aligned product system over $P$ of $C^*$-correspondences over $A$. A surjective homomorphism $\pi:{\operatorname{\mathcal{NO}}_{\!X}}\to B$ is injective if and only if $\pi$ is injective on ${\operatorname{\mathcal{NO}}_{\!X}}^\nu$ and there is a 