The squarefree twin-complete numbers conjecture

A positive integer $M$ is squarefree if it is not divisible by the square of any prime. A number is expressible as a sum of at most two positive squares but not as a sum of three positive squares if it has such a representation and no representation using three positive squares.

Squarefree twin-complete numbers conjecture. The complete list of squarefree numbers that can be written as a sum of at most two positive squares, but not as a sum of three positive squares, is

$$\{1,2,5,10,13,37,58,85,130\}.$$

By the preceding theorem, this would give the complete list of squarefree twin-complete numbers, where twin-completeness means that every vector in $\operatorname{\mathbb{Z}}^3$ of the given norm has a twin.