Bogomolov property conjecture for elliptic torsion fields

Let $K$ be a number field and let $E$ be an elliptic curve over $K$. Write $E_{\text{tor}}$ for the torsion subgroup of $E(\overline{\mathbb{Q}})$, and let $K(E_{\text{tor}})$ be the field generated over $K$ by the coordinates of the torsion points. Bogomolov property conjecture. The field $K(E_{\text{tor}})$ and the group $E(K(E_{\text{tor}}))$ have the Bogomolov property. This generalizes Habegger's theorem over $\mathbb{Q}$; the equivalent finite-extension formulation is expected to hold, while the statement is proved for elliptic curves with infinitely many supersingular primes, in particular for curves over number fields with a real embedding.