Consider the supercritical contact process on $\mathbb{Z}^2$ with infection rate $q/4$ and recovery rate $1-q$ and its space-time-diagram $\mathbb{Z}^2 \times (-\infty, 0]$. Let $\mathbb{P}_{q}$ denote the probability measure wrt to the underlying Poisson processes. For $n \in \mathbb{N}$ and $w \in \mathbb{Z}^2$ let
\begin{align*} \eta^{(n)}_w := \textbf{1}[\exists (v,t) \text{ with } ||v-w||_\infty = \lfloor \sqrt{n} \rfloor \text{ or } t = -\sqrt{n} : (v,t) \rightarrow (w, 0)] \end{align*}
and write $\eta^{(n)}_\Lambda$ for the joint distribution of the $\lbrace \eta^{(n)}_w \rbrace_{w \in \Lambda}$, $\Lambda \subseteq \mathbb{Z}^2$.
Now fix $n \in \mathbb{N}$, $b \in \lbrace 0, 1 \rbrace$ and $\Lambda = [-R,R]^2$ for some $R \in \mathbb{Z}$ and let $k \in \mathbb{N}$ be large enough such that $k \geq R+\sqrt{n}$. Dissect the subset $[-k, k]^2 \times [-k, 0]$ of the space-time-diagram canonically into stripes of length $2^{-k}$. Consider Bernoulli variables $X_1, \dots, X_K$, $K \in \mathbb{N}$ given, being $1$ either if there is a Poisson point indicating an arrow in some stripe $s_i$ or if there is no Poisson point indicating a star in $s_i$ (wlog assume each $s_i$ is on a time axis corresponding to some $v \in \Lambda$).
Further, consider the event $A := \lbrace \eta^{(n)}_\Lambda \in A' \rbrace$, $A' \subseteq \lbrace 0, 1 \rbrace ^{\Lambda}$ - for example some kind of crossing. Abbreviate $\lbrace X = b \rbrace = \lbrace X_1= \dots= X_K =b \rbrace$.
How can I bound $|\mathbb{P}_{q}(A)-\mathbb{P}_{q}(A \mid X=b)|$?
Intuitively, the influence of these few stripes should not matter, so I am quite optimistic this can be done. However, my only idea is to introduce the event
\begin{align} B &= \lbrace \text{the state of } \Lambda \text{ is independent of the Poisson points in} \cup_{i=1}^K s_i \rbrace \\ &= \lbrace \omega \in \Omega \mid \forall \omega' \in \Omega \text{ with } \omega \setminus \cup_{i=1}^K s_i = \omega' \setminus \cup_{i=1}^K s_i : \eta_\Lambda^{(n)}(\omega) = \eta_\Lambda^{(n)}(\omega') \rbrace \end{align}
because then $A \cap B$ and $\lbrace X = b \rbrace$ are independent implying
\begin{align} |\mathbb{P}_{q}(A)-\mathbb{P}_{q}(A \mid X=b)| &= |\mathbb{P}_{q}(A \cap B^c)-\mathbb{P}_{q}(A \cap B^c \mid X=b)| \\ &\leq \mathbb{P}_{q}(X=b)^{-1}|\mathbb{P}_{q}(A \cap B^c)\mathbb{P}_{q}(X=b)-\mathbb{P}_{q}(A \cap B^c)| \\ &\leq \mathbb{P}_{q}(X=b)^{-1}2\mathbb{P}_{q}(B^c) , \end{align}
which is rather unclear to me.
Remark: For my purposes it suffices to consider an up-set $A'$ resp. increasing $A$, in which case one could change $=$ to $\leq$ in the definition of $B$ - such that the state of each $1$ is independent of ... . Remark 2: I am trying to apply the ideas given in the paper of Tobias Müller on confetti percolation (2015/17) to the contact process.