Conjecture on decomposing admissible pair potentials

Let $V(|x|)$ be an admissible pair potential on $\mathbb{R}^d$. A Ruelle potential is one admitting a decomposition

$$V(|x|)=\Phi_1(|x|)+\Phi_2(|x|),$$

where $\Phi_1(|x|)$ is non-negative and tempered, while $\Phi_2(|x|)$ is stable and absolutely integrable.

Ruelle-potential decomposition conjecture. Any stable and tempered pair potential can always be written as a sum of a non-negative potential plus an absolutely summable stable potential.

The paper presents this as a weaker replacement for Ruelle's disproved conjecture: the positive-definiteness requirement on the second summand is removed. The supplied text gives no resolution of this weaker conjecture.