Completeness of maximum integral point sets over prime-power residue rings

Let $p$ be a prime and $r$ a positive integer. An integral point set in $\mathbb{Z}_{p^r}^2$ is a set of points whose pairwise squared distances are quadratic residues in $\mathbb{Z}_{p^r}$. Two such sets are considered isomorphic under the relevant affine isometries. Completeness conjecture. The above list is the complete list of maximum integral point sets in $\mathbb{Z}_{p^r}^2$ up to isomorphism. This asserts that the constructions listed in the surrounding discussion exhaust all maximum examples; no proof or resolution is supplied here.