A note on the maximum probability of ultra log-concave distributions

Jakimiuk et al. (2024) have proved that, if $X$ is an ultra log-concave random variable with integral mean, then $$\underset{n}{\max }\mathbb{P}\{X=n\}\ge \underset{n}{\max }\mathbb{P}\{Z=n\},$$ where $Z$ is a Poisson random variable with the parameter $\mathbb{E}[X]$. In this note, we show that this inequality does not always hold true when $X$ is ultra log-concave with $\mathbb{E}[X]>1$.