Lehmer-type conjecture for adelic heights above the energy minimum

Let $K$ be a number field, let $\mu$ be an adelic measure defined over $K$, let $h_\mu$ be its associated height on $\mathbb{P}^1(\overline{K})$, and let $\mathcal{L}(\mu)$ denote the energy infimum associated with $\mu$. Lehmer-type conjecture above the energy minimum. There exists $\varepsilon>0$ such that

$$[K(\alpha):K]\bigl(h_\mu(\alpha)-\mathcal{L}(\mu)\bigr)\geq\varepsilon$$

for every $\alpha\in\mathbb{P}^1(\overline{K})$ such that $h_\mu(\alpha)>\mathcal{L}(\mu)$.

This formulation is designed to remain meaningful when $\mathcal{L}(\mu)>0$. The supplied text subsequently discusses evidence that it fails for particular measures, but does not provide a resolution of the general assertion.