Uniqueness conjecture for maximally circlable sets and circles

Call a nonnegative integer $n$ maximally circlable if there exists a largest circle enclosing exactly $n$ lattice points in its interior. Fix the key triangle with vertices $(1/2,0)$, $(1/2,1/2)$, and $(1,0)$, and let $calM_n$ be a largest such circle with center in the key triangle; call it an MC-circle of $n$, and call the enclosed set of $n$ lattice points an MC-set of $n$. Rigid motions of the lattice system are translations, reflections, and rotations preserving the lattice. Uniqueness conjecture. For any maximally circlable integer $n$, both the configuration of the MC-set and the MC-circle of $n$ are unique up to rigid motions of the lattice system. The conjecture is based on overwhelming numerical evidence, but no resolution is supplied in the paper.