The normal distribution on $\mathbb{R}$, the exponential distribution on $\mathbb{R}_{\geq 0}$, and the geometric distribution on $\mathbb{N}$ are examples of distributions that are both infinitely divisible and entropy maximizers. On the other hand, the Poisson distribution is an infinitely divisible distribution on $\mathbb{N}$ without maximizing entropy, while the uniform distribution on the interval $[a,b]$ maximizes entropy but is not infinitely divisible.

Can anything be said about the relationship between these two classes of distributions?