Equality of the critical inverse temperatures for the one- and two-sided Dyson models

We prove that the critical inverse temperatures $\beta_c^{\mathbb N}(\alpha)$ and $\beta_c^{\mathbb Z}(\alpha)$ for the one- and two-sided Dyson models are the same when the power of the interaction strength $\alpha$ satisfies $1<\alpha<2$. We conjecture that this is true also in the remaining case of $\alpha=2$.