A relative Schinzel-Zassenhaus bound for the polynomial $T^2-1$

For $f(T)=T^2-1$, we establish a lower bound for the relative dynamical house of wandering algebraic integers. This bound depends on the degree over the maximal abelian extension of the rationals. Our proof follows Dimitrov’s approach to the Schinzel-Zassenhaus Conjecture and its generalization by Habegger-Schmidt.