We consider first-passage percolation on the graph $\mathbb {Z} \times \{ - h_n, -h_n+1, \ldots, h_n \}^{d-1}$ for fixed $d\ge 2$ where each edge has an i.i.d.~nonnegative weight. The passage time for a path is defined as the sum of weights of all the edges in that path and the first-passage time between two vertices is defined as the minimum passage time over all paths joining the...