A Bogomolov property for moduli spaces of polynomials over abelian extensions

Let $d\geq2$ be an integer, and let $\operatorname{MPoly}^d$ be the moduli space of degree-$d$ polynomials. For every number field $K$, we prove that there exists $ε_{K,d}>0$ such that \[ \left\lbraceα\in\operatorname{MPoly}^d(K^{\mathrm{ab}})\colon h_{\mathrm{crit}}(α)<ε_{K,d}\right\rbrace=\left\lb