  Let $V$ be an $n$-dimensional vector space over a finite field
$\mathbb{F}_q$. Define a real-valued weight function on the $1$-dimensional
vector spaces of $V$ such that the sum of all weights is zero. Let the weight
of a subspace $S$ be the sum of the weights of the $1$-dimensional subspaces
contained in $S$. In 1988 Manickam and Singhi conjectured that if $n \geq 4k$,
then the number of $k$-dimensional subspaces with nonnegative weight is at
least the number of $k$-dimensional subspaces on a fixed $1$-dimensional
subspace.
  Recently, Chowdhury, Huang, Sarkis, Shahriari, and Sudakov proved the
conjecture of Manickam and Singhi for $n \geq 3k$. We modify the technique used
by Chowdhury et al. to prove the conjecture for $n \geq 2k$ if $q$ is large.
Furthermore, if equality holds and $n \geq 2k+1$, then the set of
$k$-dimensional subspaces with nonnegative weight is the set of all
$k$-dimensional subspaces on a fixed $1$-dimensional subspace.
