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 $\alpha =2$.