The multinomial projection subgaussianity conjecture

Let $m$ and $n$ be nonnegative integers, let $S\subset\{(x_1,\dotsc,x_k)\in\mathbb Z^n:x_i\geq0,\ x_1+\dotsb+x_k=m\}$, and let $(p_1,\dotsc,p_k)\sim\operatorname{Dir}(\alpha_1,\dotsc,\alpha_k)$. The associated multinomial probability of $S$ is

$$\sum_{(x_1,\dotsc,x_k)\in S}\frac{m!}{x_1!\dotsm x_k!}p_1^{x_1}\dotsm p_k^{x_k}.$$

Multinomial projection subgaussianity conjecture. This random variable is $O\left(\frac{m}{\alpha_1+\dotsm+\alpha_k}\right)$-subgaussian. The conjecture is part of a family of concentration conditions for conjugate-prior models in adaptive data analysis; the supplied text does not indicate whether it has been resolved.