Let us suppose that we are in the setting of Janson's inequality for Poisson-type deviations of increasing events. Specifically, we have independent Bernoulli variables $X_1, \dots, X_n$, and events $A_1, \dots, A_m$, each of which is an increasing function of some subset $S_i$ of the $X$ variables. For the purposes of this question, we can in fact assume that each event $A_i$ has the form $A_i = \bigwedge_{k \in S_i} X_k$.

We write $i \sim j$ if $i \neq j$ and $S_i \cap S_j \neq \emptyset$. Define $\mu = \sum_i \Pr(A_i)$ and $\Delta = \sum_{i,j: i \sim j} \Pr(A_i \cap A_j)$.

Let us define the event $$ B = \bigvee_{i=1}^m \Bigl( A_i \cap \bigwedge_{j: j \sim i} \overline A_j \Bigr), $$ i.e. that some event $A_i$ holds, but none of its neighbors $A_j$ do. I would like to show a lower bound on $\Pr(B)$, assuming that $\mu \gg \Delta$.

We can count the expected number of indices $i$ with this property; by the union bound, it is at least $\sum_{i=1}^m \Pr(A_i) \Bigl(1 - \sum_{j: j \sim i} \Pr(A_i \cap A_j) \Bigr) = \mu - \Delta$. My question is whether a Janson-type bound holds in this setting. For example, I might conjecture that $$ \Pr(B) \geq 1 - e^{-\mu + \Delta} $$

Is such a bound possible to show?