Heavy-tail conjecture for unbounded signal power in SIR models

Let $x_0$ be the desired transmitter, let $\ell(x_0)$ be its path-loss term, let $P_{\rm s}(\theta)$ denote the success probability at SIR threshold $\theta$, and let $\delta$ be the path-loss parameter. The reciprocal $1/\ell(x_0)$ describes the inverse mean signal power.

Heavy-tail conjecture.

$$0\in\operatorname{supp}(1/\ell(x_0)) \quad \Longleftrightarrow \quad P_{\rm s}(\theta)=\Theta(\theta^{-\delta}),\quad\theta\to\infty.$$

The claim predicts a power-law upper tail with exponent $\delta$ exactly when the mean signal power can become arbitrarily large. The source states it as one of three conjectures summarizing its cellular and ad hoc asymptotic results; its resolution is not specified.