= Solution
Use integer <graph vertices> in the boxes: $\Lambda_n=[-n,n]^2\cap\mathbb Z^2$, and set $\Lambda_{-1}=\varnothing$. Thus $\partial\Lambda_0=\{0\}$, $g_0=1$, and $|\partial\Lambda_m|=8m$ for $m\geq1$.
If $0\leftrightarrow\partial\Lambda_{m+n}$, erase loops from an open <graph path> to obtain a <self-avoiding walk>. Let $x$ be its first <graph vertex> on $\partial\Lambda_m$. Split at $x$. The prefix certifies $A_x=\{0\leftrightarrow x\}$, while a segment of the suffix certifies $B_x=\{x\leftrightarrow x+\partial\Lambda_n\}$: the final <graph vertex> is at <supremum norm> distance at least $n$ from $x$, so the suffix first hits that translated boundary. The two certificates use disjoint <edges>. Therefore the <BK boundary-splitting estimate>, the <union bound>, the <Van den Berg-Kesten inequality>, and translation invariance give
$$
\begin{aligned}
g_{m+n}&\leq\sum_{x\in\partial\Lambda_m}\mathbb P_p(A_x\mathbin\square B_x)\\
&\leq\sum_{x\in\partial\Lambda_m}\mathbb P_p(A_x)\mathbb P_p(B_x)\\
&\leq|\partial\Lambda_m|g_mg_n.
\end{aligned}
$$
The cases $m=0$ or $n=0$ follow directly from $g_0=1$, so this proves the bound including the endpoints.
Here is an explicit way to remove the polynomial boundary factor. For $p>0$ put $b_n=32n^2g_n$, $n\geq1$. By symmetry in $m,n$, assume $m\leq n$. Then
$$
\frac{b_{m+n}}{b_mb_n}\leq\frac{(m+n)^2}{4mn^2}\leq\frac1m\leq1.
$$
Consequently $a_n=\log b_n$ is a <subadditive sequence>. The <Fekete lemma> states that any real <subadditive sequence> satisfies $\lim a_n/n=\inf_{n\geq1}a_n/n$, possibly $-\infty$. In this case the direct horizontal open <graph path> gives $g_n\geq p^n$, so this <limit of a sequence> is bounded below by $\log p$. Since $g_n\leq1$, its upper bound is zero. Finally,
$$
\frac{\log g_n}{n}=\frac{a_n}{n}-\frac{\log(32n^2)}n
$$
has the same <limit of a sequence>. Thus the <percolation one-arm decay rate> exists and
$$
\boxed{\gamma=\lim_{n\to\infty}g_n^{1/n}\in[p,1]}.
$$
For $p=0$, every $g_n$ with $n\geq1$ is zero and $\gamma=0$. No exponential-decay theorem or assumption that $p$ is below the <percolation critical probability> is needed.
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