Poisson condition $(S_2)$ conjecture

Let $m$ be a positive integer, and let $X$ be a Poisson random variable with parameter $m$:

$$X\sim\operatorname{Poi}(m).$$

Poisson $(S_2)$ conjecture. $X$ satisfies Condition $(S_2)$.

If true, this statement, together with the paper's general theorem, would provide another proof of the paper's main Poisson inequality. The source presents it as a statement whose correctness is not established.