The property in question is a special case (with the probabilities of all the $k$ outcomes equal to one another) of the known $NA$ (negative association) property of the multinomial distribution; see e.g. [this sentence in the bottom paragraph on page 5][1]: 

>$NA$ property of multinomial distributions can be seen from Condition $N$, since it is the conditional distribution of independent Poisson random variables given their sum.


  [1]: https://arxiv.org/abs/1803.09663v1