= Solution
Under <plus boundary condition>[plus boundary conditions], if $\sigma_0=-1$, the negative cluster containing the origin is surrounded by a <Peierls contour> $\gamma$. Flipping all spins inside $\gamma$ is injective after $\gamma$ is specified and increases the Boltzmann weight by $e^{2\beta|\gamma|}$. Therefore
$$
\mathbb P_{\Lambda_L,\beta}^{+}(\sigma_0=-1)
\leq\sum_{\gamma\ni0}e^{-2\beta|\gamma|}.
$$
The number of length-$m$ contours surrounding the origin is at most $C m3^m$, so the sum tends to zero as $\beta\to\infty$, uniformly in $L$. For sufficiently large $\beta$ it is below $1/4$, and then
$$
\langle\sigma_0\rangle_{\Lambda_L,\beta}^{+}
=1-2\mathbb P(\sigma_0=-1)\geq\frac12.
$$
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