Exponential-tail conjecture for bounded signal power with Rayleigh fading

Let $x_0$ be the desired transmitter, let $\ell(x_0)$ be its path-loss term, and let $P_{\rm s}(\theta)$ denote the success probability at SIR threshold $\theta$. Assume Rayleigh fading, and let $\delta$ be the path-loss parameter.

Exponential-tail conjecture. For Rayleigh fading,

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

The conjecture predicts exponential upper-tail decay when the signal power is bounded under Rayleigh fading. The paper motivates it from exponential-tail results in bounded-path-loss cellular and ad hoc models, but does not establish the general equivalence.