  In 2013, Koldobsky posed the problem to find a constant $d_n$, depending only
on the dimension $n$, such that for any origin-symmetric convex body
$K\subset\mathbb{R}^n$ there exists an $(n-1)$-dimensional linear subspace
$H\subset\mathbb{R}^n$ with \[
  |K\cap\mathbb Z^n| \leq d_n\,|K\cap H\cap \mathbb
Z^n|\,\mathrm{vol}(K)^{\frac 1n}. \] In this article we show that $d_n$ is
bounded from above by $c\,n^2\,\omega(n)/\log(n)$, where $c$ is an absolute
constant and $\omega(n)$ is the flatness constant. Due to the recent best known
upper bound on $\omega(n)$ we get a ${c\,n^3\log(n)^2}$ bound on $d_n$. This
improves on former bounds which were exponential in the dimension.
