The conjectural formula for β(N)

Let $N\geq 2$ be an integer, and write $\beta(N)=\beta(N,N)$. Formula conjecture.

1. If $N\not\equiv 1,9\pmod{24}$, then $\beta(N)$ is periodic in $N$ with period $24$, with values given by the paper's table.
2. If $N\equiv 9\pmod{24}$, then $\beta(N)=8$ when $N$ is a perfect square and $\beta(N)=4$ otherwise.
3. If $N\equiv 1\pmod{24}$, then $\beta(N)=24$ when $N$ is a perfect square and $\beta(N)=12$ otherwise.

The first part was verified for $N\leq 960$, and the second and third parts for $N<3000$, by a SageMath program. The conjecture gives an explicit description of the image invariant $\beta(N)$, but remains unproved in the source.