I want to know whether  the following version of GKS inequality with boundary condition for Ising model hold or not. Consider Ising model on $\mathbb{Z}^d$ and $\varnothing \neq A\subset \Lambda_1 \subset \Lambda_2 \subset \mathbb{Z}^d$. Let $x \in \Lambda_1 \setminus A$ and $\eta_A$ be a fixed configuration on $A$. Is the following true?
\begin{equation} 
\langle \sigma_x \rangle_{\Lambda_1}^{\eta_A} \leq \langle \sigma_x \rangle_{\Lambda_2}^{\eta_A}, \qquad (\star)
\end{equation}
where $\langle \cdot \rangle_{\Lambda}^{\eta}$ is the expectation w.r.t to Gibbs measure on $\Lambda$ with boundary condition $\eta$. 

I know that the original GKS inequality works with free boundary condition, i.e. $A=\varnothing$, and implies that for all $B \subset \Lambda_1$ we have $\langle  \sigma_B \rangle_{\Lambda_1} \leq \langle \sigma_B \rangle_{\Lambda_2}$, with $\sigma_B=\prod_{y\in B} \sigma_y$.

So, can we prove or disprove the inequality  $(\star)$?

Thank you.