= Solution
Couple the free-boundary random-cluster heat-bath chains in $\Lambda_N$ and $\Lambda_{N+1}$. Restricted to common edges, the larger box has extra possible open paths through its outer annulus. These can only turn a disconnected-endpoint update into a connected-endpoint update and raise its open probability from $p/(2-p)$ to $p$. The monotone coupling therefore gives
$$
P_N\preceq P_{N+1}
$$
on common increasing events. Consequently $P_N(A)$ is nondecreasing and, being bounded by one, has a limit.
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