An analogue of the Aleksandrov projection theorem for convex lattice polygons

Let $K$ and $L$ be origin-symmetric convex lattice polytopes in $\mathbb {R}^n$. We study a discrete analogue of the Aleksandrov projection theorem. If for every $u\in \mathbb {Z}^n$, the sets $(K\cap \mathbb {Z}^n)|u^\perp$ and $(L\cap \mathbb {Z}^n)|u^\perp$ have the same number of points, is $K=L$? We give a positive answer to this problem in $\mathbb {Z}^2$ under the additional hypothesis that $(2K\cap \mathbb {Z}^2)|u^\perp$ and $(2L\cap \mathbb {Z}^2)|u^\perp$ have the same number of points for every $u\in \mathbb {Z}^n$.