Supersingular primes and Bogomolov property

Let $E$ be an elliptic curve over a number field $K$ with at least one real embedding and $L$ be a finite extension of $K$. We generalize a result of Habegger to show that $L(E_{\text{tor}})$, the field generated by the torsion points of $E$ over $L$, has the Bogomolov property. Moreover, the Néron-Tate height on $E(L(E_{\text{tor}}))$ also satisfies a similar discreteness property. Our main tool is a general criterion of Plessis that reduces the problem to the existence of a supersingular prime for $E$ satisfying certain conditions.