Proposition: Finite records do not bound unobserved energy

Fix any state ${\rho }_S$ and any family of measurements acting only on $S$. Append an unobserved oscillator with Hamiltonian $H_O=\omega N$, $\omega >0$. The states

give identical records for every such measurement, while their unobserved energy is $n\omega$.

Proof. The measurements act as the identity on $O$, so their probabilities depend only on ${\rho }_S$. The energy follows from $N|n⟩=n|n⟩$.