Consider a Hamiltonian $H$ on some spin chain of length $L$.
Suppose we have a subset of $n$ eigenstates $\{|\psi_i\rangle \}$ of $H$ obeying the following special condition. First, a couple quick definitions: $$|0, \{a_i\}\rangle = \sum_{i=1}^n a_i|\psi_i\rangle,$$ $$|t, \{a_i\}\rangle = e^{-iHt}|0, \{a_i\}\rangle$$
The special condition is that for these special eigenstates of $H$, every time $t$, and every choice of $\{a_i \}$, we can find a product of (potentially $t$ and $\{a_i\}$-dependent) single-site unitaries such that:
$$|t, \{a_i\}\rangle = \left( \bigotimes_{j=1}^L U_j(t, \{a_i\})\right)|0, \{a_i\}\rangle$$
This condition means that the value of any entanglement measure of this superposition of states cannot change in time.
If this special condition is met, can we additionally find a set of Hermitian single-site operators $O_j$ such that
$$|t, \{a_i\}\rangle = \left( \bigotimes_{j=1}^L e^{-i \sum_{j=1}^L O_j t}\right)|0, \{a_i\}\rangle?$$
Here the $O_j$ are independent of $t$ and $\{a_j\}$.
The idea very roughly is the following. I want to use the fact that our special condition is met for many states in order to argue that we can find $U_j$ that work for all $\{a_j\}$. I want to use the fact that these states have a special $t$-dependence to argue that we can find $U_j$ with a similarly simple $t$ dependence stemming from a time-independent Hamiltonian.
