Manickam–Miklós–Singhi conjecture

Let $n,k$ be natural numbers with $n\geq 4k$, and let $x_1,x_2,\dots,x_n$ be real numbers satisfying

$$\sum_{i=1}^n x_i\geq 0.$$

Manickam–Miklós–Singhi conjecture. There exist at least $\binom{n-1}{k-1}$ subsets $A\subseteq [n]$ such that $|A|=k$ and

$$\sum_{i\in A}x_i\geq 0.$$

The bound is motivated by assigning one element weight $1$ and each of the remaining $n-1$ elements weight $-1/(n-1)$, for which exactly the $k$-subsets containing the positive element have nonnegative sum. The source presents this as the MMS conjecture in the range $n\geq 4k$; no resolution evidence is supplied here.