Amoroso–Zannier's relative Lehmer conjecture

Let $K$ be a number field, let $\mathbb{G}_m$ denote the multiplicative group, and let $\mu_\infty$ denote the roots of unity. For $x\in\mathbb{G}_m(\overline{K})\setminus\mu_\infty$, set

$$D=[K^{\mathrm{ab}}(x):K^{\mathrm{ab}}].$$

Amoroso–Zannier's conjecture. There exists a constant $c(K)>0$ such that

$$h(x)\geq\frac{c(K)}{D}.$$

This is the relative abelian extension analogue of Lehmer's conjecture. The cited work establishes a lower bound with an additional logarithmic factor, while the conjectural bound itself remains open.