= Solution
The <union bound> gives $g_k\leq\sum_{x\in\partial\Lambda_k}\mathbb P_p(0\leftrightarrow x)$. Hence some boundary <graph vertex> has <percolation two-point connection probability> at least $g_k/(8k)$. The rotations and reflections of the <square lattice> let us choose such a <graph vertex> as $x=(k,j)$, $-k\leq j\leq k$. Reflection in the vertical line through $x$ sends $0$ to $e_{2k}=(2k,0)$ and fixes $x$. Therefore
$$
\mathbb P_p(x\leftrightarrow e_{2k})=\mathbb P_p(0\leftrightarrow x).
$$
The <Harris-FKG inequality> applied to these two <increasing events> proves the <reflection lower bound for two-point percolation>:
$$
h_{2k}\geq\mathbb P_p(0\leftrightarrow x,\ x\leftrightarrow e_{2k})\geq\left(\frac{g_k}{8k}\right)^2.
$$
Every <graph path> from $0$ to $e_{2k}$ reaches $\partial\Lambda_{2k}$, so $h_{2k}\leq g_{2k}$. Taking roots gives
$$
\left(\frac{g_k}{8k}\right)^{1/k}\leq h_{2k}^{1/(2k)}\leq g_{2k}^{1/(2k)}.
$$
Both outside expressions have <limit of a sequence> $\gamma$, proving the even case. For $p>0$, the <Harris-FKG inequality> with the last horizontal <edge> gives $h_{2k+1}\geq p h_{2k}$, and also $h_{2k+1}\leq g_{2k+1}$. The lower bound has root
$$
(p h_{2k})^{1/(2k+1)}=p^{1/(2k+1)}\left(h_{2k}^{1/(2k)}\right)^{2k/(2k+1)}\longrightarrow\gamma.
$$
The upper bound has the same <limit of a sequence>. At $p=0$ all positive-distance connection <probabilities> vanish. \b[Thus $\lim_{n\to\infty}h_n^{1/n}=\gamma$ for every $p\in[0,1]$.]
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