The line-intersection bound conjecture for cyclic tilings

Let $A\subset\mathbb{Z}_M$, where $M=\prod_{j=1}^K p_j^{n_j}$, and suppose that $p_i^{\alpha_i}\parallel |A|$ with $\alpha_i<n_i$. Let $\ell_i(x)$ denote the line in the $p_i$ direction through $x$.

Line-intersection bound conjecture. For every $x\in\mathbb{Z}_M$,

$$|A\cap\ell_i(x)|<p_i^{n_i}.$$

The conjecture is a proposed lower-dimensional analogue of the subspace bounds used in the paper. It is stated as a modest conjecture, and the supplied text gives no resolution.