Asymptotically positive extensions and property (B)

Let $\mathcal{L}/\mathbb{Q}$ be an infinite extension. For a prime power $q$, let $\psi_q$ denote the limiting normalized number of prime ideals of norm $q$ in finite subextensions of $\mathcal{L}$, and let $\mathcal{K}$ denote the set of nonzero elements of $\mathcal{L}$ that are not roots of unity. The extension is asymptotically positive if $\psi_q>0$ for some prime power $q$. Asymptotically positive extension conjecture.

$$\liminf_{\alpha\in\mathcal{K}}h(\alpha)\geq\frac{1}{2}\sum_q\psi_q\frac{\log q}{q+1},$$

where $q$ runs over all prime powers. Consequently, every asymptotically positive extension $\mathcal{L}$ has property (B). This was proposed in earlier work and is presented here as an open conjecture; property (B) asserts a positive lower bound for the height of non-torsion elements.