Quasi-local Araki conjecture

Let $Q_\Sigma = \sum_{Z\subset \Sigma} q_Z$ be a quasi-local Hamiltonian on $\Sigma$ with strength $J$, and let $f$ be a quasi-local observable with center at the origin. Let $Q_{\ell} = \sum_{Z\subset [-\ell, \ell]} q_Z$. For $\tau=2Jt$, there should exist analytic functions $G$ and $H$, independent of the length of $\Sigma$, such that

$$\left\|\Gamma^t_\Sigma(f)-\Gamma^t_\ell(f)\right\| \leq \mu_{\ell}(\tau)G(\tau)\left\|f\right\|$$

and

$$\left\|\Gamma^t_\Sigma(f)\right\| \leq H(\tau)\left\|f\right\|.$$

Quasi-local Araki conjecture. The displayed approximation and norm bounds hold for quasi-local Hamiltonian interactions.

This would extend Araki's theorem from local to quasi-local interactions and provide the estimates needed to extend the paper's locality arguments. The source presents it as an expected extension rather than an established result, and does not provide a proof.