The Unruh effect is a well-known example in which two Hamiltonians $H$ and $\hat H$ associated with different timelike Killing vector fields both have a lower bound, in the same Hilbert-space representation, even though they are not related to each other by any spacetime isometry. This question asks about a generalization.
Consider a quantum field theory in flat spacetime, expressed in terms of field operators acting on a Hilbert space. Let $K$ and $\hat K$ be two different timelike Killing vector fields, not necessarily related to each other by any isometry, and not necessarily covering the whole spacetime. (As an example, think of Rindler coordinates.) Let $R$ be the region of spacetime in which both Killing vector fields are defined, and consider the algebra of observables in $R$. Let $H$ and $\hat H$ be the operators (Hamiltonians) that generate translations of these observables along $K$ and $\hat K$, respectively.
Question: Suppose that the algebra is represented on a Hilbert space in such a way that the spectrum of one of the Hamiltonians $H$ has a lower bound. Does this imply that the spectrum of the other Hamiltonian $\hat H$ also has a lower bound (in the same Hilbert-space representation)?$^\dagger$
I'm not looking for a watertight proof, just a compelling argument — something clear enough that I could check each step in a free field theory.
By the way, in case this isn't familiar: the Hamiltonian density is not necessarily positive definite in quantum field theory, not even in a representation where the Hamiltonian itself is positive definite. See Fewster (2005) "Energy Inequalities in Quantum Field Theory", https://arxiv.org/abs/math-ph/0501073, which says (page 2):
quantum fields have long been known to violate all such pointwise energy conditions  and, in many models, the energy density is in fact unbounded from below on the class of physically reasonable states.
$^\dagger$ The question refers to how the operators are represented on a Hilbert space. That's important because $H$ typically does not have a lower bound in most Hilbert-space representations even if it does in one of them. The spectrum condition is a property of a specific Hilbert-space representation, not just a property of the abstract algebra of observables.