= Solution
On $A_N$, additionally require every site of $h_{N+1}$ to be open and every site of $h_{N+2}$ to be closed. The open ring joins every component of $C_N$ to the origin component, and the closed outer ring makes that component finite. It contains at least $\theta\#H_N/2$ sites. These ring states are independent of $A_N$, and their probability is
$$
p^{\#h_{N+1}}(1-p)^{\#h_{N+2}}.
$$
Since both ring sizes are linear in $N$, this is at least $c(p)e^{-u(p)N}$. Multiplication by $\mathbb P(A_N)\geq\theta^2/2$ proves the claimed lower bound.
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