It seems pretty clear that if you take a very diluted subset of, say, the horizontal line through $0$, then you'll be able to make a Peierls argument.
For example, put $h=+\infty$ (worst possible case, amounting to fixing the corresponding spins to $+1$) at all vertices with coordinates (10^k,0), with $k\geq 1$. Then, when removing a contour surrounding a given site $i$, i.e. flipping all spins inside the contour, except for the frozen ones, we gain an energy proportional to the length of the contour, and only lose an energy at worst proportional to the number of frozen spins surrounded by the contour. The latter term is always much smaller than the former one (at least if $i$ is taken far enough from the frozen spins).
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