A plane lattice $\Lambda$ is a set $\Lambda= \{ mA+nB: m,n \in \mathbb Z \}$, where $A,B$ are linearly independent vectors in $\mathbb R^2$. The set of all circles in $\Lambda$ is
$$\mathcal K(\Lambda) = \Bigl\{\bigl\{X \in \Lambda:\|X-C\|=R\bigr\} : C \in \mathbb R^2 , R \in \mathbb R_{\ge 0} \Bigr\},$$
where $\|\cdot\|$ is the usual Euclidean metric. Let's define the function $\psi_\Lambda$, for every plane lattice $\Lambda$, \begin{align} \psi_{\Lambda}\colon \mathcal K(\Lambda) &\to \mathbb N\\ \gamma &\mapsto |\gamma|. \end{align}
Conjecture. There is a plane lattice $\Lambda$ such that $\psi_\Lambda$ is bounded.
I found some lattices $\Lambda$ whose circles with centres in $\Lambda$ have at most two points. Is my conjecture true in the general case?
Update. This question has been answered on mathoverflow.