= Solution
For each of the $2d$ coordinate faces $F$ of $B(N)$, define the <increasing event> $A_F(m,N)$ that an open <path in a graph> lying in $B(N)$ joins $B(m)$ to $F$. Their union is exactly the event of connection from $B(m)$ to $\partial B(N)$ within $B(N)$. The <bond percolation> law and the centered boxes are invariant under coordinate permutations and sign changes, so all $2d$ events have the same probability.
Apply the <square-root trick for positively associated events> from part (a), with $2d$ in place of the number of events. If $\delta_m=1-\mathbb P_p(D_m)$, part (b.i) gives, for every $N=m+n$,
$$
\boxed{\mathbb P_p(A_F(m,N))\geq1-\delta_m^{1/(2d)}\longrightarrow1,\qquad\text{uniformly in }n\geq1.}
$$
In particular this holds for the left and right faces, and it proves the stronger statement in which each connecting <path in a graph> stays inside the outer box.
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