Binomial condition $(S_2)$ conjecture

Let $n$ and $m$ be positive integers with $n>2m$, and let $X$ be a binomial random variable with parameters $n$ and $\frac{m}{n}$:

$$X\sim\operatorname{Bin}\left(n,\frac{m}{n}\right).$$

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

If true, this would extend the range in which the paper's general theorem yields the Jogdeo–Samuels inequality, and may be viewed as a refinement of an older result of Simmons. The authors state that they are unable to verify the condition but believe it holds.