= Solution
Write $\phi=\phi^1_{p,q}$ and $C_n=\phi(0\leftrightarrow e_n)$. We use these standard facts about the <infinite-volume wired random-cluster measure> for $q\ge1$: it is translation invariant and invariant under coordinate reflections/permutations; it has <positive association of the random-cluster model>; and every conditional edge-open <probability> is at least
$$
r=\frac{p}{p+q(1-p)}>0.
$$
The last bound follows because an <edge> either leaves its component count unchanged on opening, giving <probability> $p$, or merges two components, giving $p/(p+q(1-p))$. These are standard infinite-volume properties of the limit obtained from the finite wired laws by <weak convergence of probability measures>. <Positive association of random variables> applies to connection events by increasing approximation with finite-path events.
Opening the $n$ <edges> of the straight segment gives $C_n\ge r^n$, so its logarithm is finite. The events $A=\{0\leftrightarrow e_m\}$ and $B=\{e_m\leftrightarrow e_{m+n}\}$ are increasing, and their intersection implies connection to $e_{m+n}$. Thus
$$
C_{m+n}\ge\phi(A\cap B)\ge\phi(A)\phi(B)=C_mC_n.
$$
The sequence $a_n=-\log C_n$ is a <subadditive sequence>, with $0\le a_n\le-n\log r$. Part (a) proves the <random-cluster inverse correlation length> exists and is finite:
$$
\boxed{\alpha(p,q)=\lim_n\frac{a_n}{n}=\inf_n\frac{a_n}{n}\in[0,-\log r].}
$$
Since $a_n/n\ge\alpha$ for every $n$, exponentiating yields the requested bound
$$
\boxed{\phi^1_{p,q}(0\leftrightarrow e_n)\le e^{-n\alpha(p,q)}.}
$$
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