Example: A Hamiltonian tail is not an exact light cone

For two assigned separated sites, take $H=J{X}_A{X}_B$, $J>0$. Prepare $B$ in $|0⟩$, and replace $A$ by $|+⟩$ or $|-⟩$. Their output states on $B$ are $e^{\mp iJtX/\hslash }|0⟩$, with trace distance

It is nonzero for arbitrarily small positive $t$. Such a Hamiltonian cannot be identified with an exact geometrically local communication primitive at positive separation.