In their famed paper (https://arxiv.org/abs/math-ph/0011053), Bourgain and Goldstein conjecture what they call the paving property:
Let $H_{jk}=\delta_{j,k+1}+\delta_{j,k-1}+v(\theta+j\omega)\delta_{jk}$ be the matrix of the Schrödinger operator $H$ for a quasi-periodic nearest-neighbour lattice system with $j,k\in\mathbb Z$ and $\theta,\omega\in\mathbb T$.
Denote by $H_\Delta$ the restriction of $H$ to $\Delta\subset\mathbb Z$, i.e. $H_\Delta=P_\Delta HP_\Delta$ with $P_\Delta$ the projection onto $\Delta$, and by $G_\Delta=(H_\Delta-E)^{-1}$ its resolvent for $E\notin\sigma(H_\Delta)$.
Then: let $\Delta\subset\mathbb Z$ be an interval of length $N>n$ such that for each $x\in\Delta$ there is an interval $\Delta'\subset\Delta$ of length $n$ satisfying \begin{equation*} \{y\in\Delta:|x-y|<\frac n{10}\}\subset\Delta' \end{equation*} and \begin{equation*} |G_{\Delta'}(n_1,n_2)|<e^{-c|n_1-n_2|+o(n)} \end{equation*} for some constant $c>0$ and $n$ sufficiently large. Then also \begin{equation*} |G_{\Delta}(n_1,n_2)|<e^{-\frac c2|n_1-n_2|+o(n)}. \end{equation*} The authors promise a proof of this statement in Section IV of their paper which does not exist.
Is the statement obvious and if so, why and how?