The circle conjecture for plane integral point sets in semi-general position

A plane integral point set is a finite set of points in the Euclidean plane whose pairwise distances are integers; it is in semi-general position when no three points are collinear. A set has minimum diameter when its diameter is minimal among plane integral point sets of the same cardinality.

Circle conjecture. The points of a plane integral point set in semi-general position are situated on a circle.

The paper notes that all known examples in semi-general position with minimum diameter lie on a common circle, and proves this structure for sets with at most $36$ points. The conjecture asserts the circle property in general.