Given a (non-unital) $C^*$-algebra $A$, I'll call a state $\phi$ on $A$ compactly supported if $\phi$ attains its norm on $A$. For example, every pure state is compactly supported (which follows from Kadison's transitivity theorem).
Is it true/known that the adjoint of a proper $*$-homomorphism $f:A \to B$ takes compactly supported states on $B$ to compactly supported states on $A$? Note: a $*$-homomorphism is proper if it sends approximate identities to approximate identities, so that the adjoint at least preserves the set of states.
If it is known, could I please have a reference to proof/counterexample? If unknown, I would really appreciate any references which might lead towards some progress.