are elementary symmetric polynomials concave on probability distributions?

Question

Let $S_{n,k}=\sum_{S\subset[n],|S|=k}\prod_{i\in S} x_i$ be the elementary symmetric polynomial of degree $k$ on $n$ variables. Consider this polynomial as a function, in particular a function on probability distributions on $n$ items. It is not hard to see that this function is maximized at the uniform distribution. I am wondering if there is a "convexity"-based approach to show this. Specifically, is $S_{n,k}$ concave on probability distributions on $n$ items?