The product conjecture for fields with the Bogomolov property

Let $E$ and $F$ be fields of algebraic numbers. A set has the Bogomolov property if there exists a positive constant $c$ such that the Weil height of each element is either $0$ or at least $c$. The group product $E^*F^*$ consists of products $ef$ with $e\in E^*$ and $f\in F^*$.

Product conjecture. If $E$ and $F$ have the Bogomolov property, then the group product $E^*F^*$ also has the Bogomolov property.

The statement is suggested by the preceding example and is presented as a conjectural strengthening of the established result for $({\mathbb Q}^{\rm ab})^*({\mathbb Q}^{\rm tr})^*$. The source indicates that an even more general result will be proved later, so the conjectural statement is expected to be resolved within the paper.