Lehmer-type conjecture for adelic heights

Let $K$ be a number field, let $\mu$ be an adelic measure defined over $K$, and let $h_\mu$ be its associated adelic height on $\mathbb{P}^1(\overline{K})$. Lehmer-type conjecture for adelic heights. There exists $\varepsilon>0$ such that

$$[K(\alpha):K]h_\mu(\alpha)\geq\varepsilon$$

for every $\alpha\in\mathbb{P}^1(\overline{K})$ with $h_\mu(\alpha)>0$.

The paper notes that this formulation is trivially true when the energy infimum $\mathcal{L}(\mu)$ is positive, so it is not the appropriate generalization for the measures considered later. Its general status is not resolved in the supplied text.