An entropic analogue of the MMS conjecture

Let $P=\{x_1,\dots ,x_n\}$ be a multiset consisting of $n\ge 2$ real numbers such that ${\sum }_{i=1}^n{x}_i=0$ and ${\sum }_{i=1}^n\mathrm{\mid }{x}_i\mathrm{\mid }>0$, and let $k<n$ be a positive integer. We sample $k$ elements from $P$ without replacement and set $X_P$ be the sum of the elements in our sample. It is shown that the Shannon entropy of $X_P$ satisfies $$\mathbf{H}(X_P)\ge \mathbf{H}(\text{Ber}(k\mathrm{/}n)),$$ where $\text{Ber}(k\mathrm{/}n)$ is a Bernoulli random variable of mean $k\mathrm{/}n$. The result is sharp, and may be seen as an entropic analogue of the Manickam-Miklós-Singhi (MMS) conjecture.