GKS inequality with boundary condition 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.
Update after the comment by Prof. Yvan Velenik.
As shown by Yvan, the inequality $(\star)$ is not correct in general setting.
I am working with a particular setting as follows. Consider a connected graph $G$ containing several components: $G=A\cup B \cup C$, where $V(A)\cap V(B)=\varnothing$, there is only one edge, say $\{x,y\}$ between $A$ and $B$ with $y \in A$ and $x \in B$;   and $C=D \cup E$ with $D\cap A = \varnothing$, $E\cap B=\varnothing$ and $B$ is connected to $D$ and $E$ is connected to both $A$ and $D$. The question is still to check
$$\langle \sigma_x \rangle^{\eta_A}_{B} \leq \langle \sigma_x \rangle^{\eta_A}_G ?  \qquad (1)$$
I'am sorry for the complicated construction of $G$. Actually, I'am considering the case $G$ is a random graph locally tree like. I want to truncate the Gibbs measure on $G$ to the measure on a ball (say $B$) around the vertex $x$. So we can expect $A$ and $B$ are very near to trees (there are  only a few cycles in $A\cup B$). Assume $|G|=n$ and $|A\cup B|$ is more ore less $(\log n)^3$.
If the inequality is not true, can we expect an approximation like
$$\langle \sigma_x \rangle^{\eta_A}_{B} \leq \langle \sigma_x \rangle^{\eta_A}_G  +  o_n(1)? \qquad (2)$$
where $o_n(1)$ depends only on $n$.
Thank you.
 A: You can consider the measures in $\Lambda_1\setminus A$ and $\Lambda_2\setminus A$ with free boundary condition, replacing the effect of $\eta_A$ by a suitable magnetic field acting on the vertices on the exterior boundary of $A$.
If this induced magnetic field is nonnegative, then your inequality $(\star)$ is just GKS. This is in particular the case if $\eta_A$ has only $+$ spins along the inner boundary of $A$.
In general, however, the inequality $(\star)$ does not hold. If it did hold for general boundary conditions $\eta$, then you'd get absurd results. Indeed, consider the Ising model without magnetic field. Then,
$$
\langle\sigma_x\rangle_{\Lambda_1}^{\eta_A} = -\langle\sigma_x\rangle_{\Lambda_1}^{-\eta_A} \geq -\langle\sigma_x\rangle_{\Lambda_2}^{-\eta_A} =
 \langle\sigma_x\rangle_{\Lambda_2}^{\eta_A},
$$
where the two equalities follow the spin-flip symmetry, and the inequality from $(\star)$. Since, by $(\star)$, you'd also have
$$
\langle\sigma_x\rangle_{\Lambda_1}^{\eta_A} \leq
\langle\sigma_x\rangle_{\Lambda_2}^{\eta_A},
$$
one would deduce that $\langle\sigma_x\rangle_{\Lambda_1}^{\eta_A} =
\langle\sigma_x\rangle_{\Lambda_2}^{\eta_A}$, which is easily seen to be false in general.

What I mean by "replacing the effect of $\eta_A$ by a suitable magnetic field acting on the vertices on the exterior boundary of $A$" is the following: the energy in $\Lambda_1\setminus A$, given that there is a boundary condition $\eta_A$ on $A$, is
$$
- \sum_{\{i,j\}\subset\Lambda_1\setminus A} J_{i,j}\sigma_i\sigma_j - \sum_{i\in \Lambda_1\setminus A}\sum_{j\in A} J_{i,j}\sigma_i\eta_j
=
- \sum_{\{i,j\}\subset\Lambda_1\setminus A} J_{i,j}\sigma_i\sigma_j - \sum_{i\in \Lambda_1\setminus A} \Bigl(\sum_{j\in A} J_{i,j}\eta_j \Bigr) \sigma_i
,
$$
so that the effect of the boundary condition can be seen as effective magnetic fields $h_i$ acting on vertices $i\in\Lambda_1\setminus A$ and given by
$$
h_i = \sum_{j\in A} J_{i,j}\eta_j.
$$
