= Solution
Let $A_n$ be the increasing cylinder event that an open path joins $0$ to the boundary of the box $[-n,n]^d$. If two edge-disjoint open paths run from $0$ to infinity, then $A_n\mathbin\square A_n$ occurs for every $n$. Therefore the <van den Berg-Kesten inequality> gives
$$
\mathbb P_p(\text{two edge-disjoint open paths }0\leftrightarrow\infty)
\leq\lim_{n\to\infty}\mathbb P_p(A_n)^2
=\theta(p)^2,
$$
where continuity from above identifies $\lim_n\mathbb P_p(A_n)=\theta(p)$.
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