Non-periodic connection points on double regular n-gons for odd n at least 9

Let $n\geq 9$ be odd, and consider the double regular $n$-gon translation surface. A non-periodic connection point is a connection point that is not periodic under the affine automorphism group.

Non-periodic connection-point conjecture. For $n\geq 9$ odd, there are no non-periodic connection points on the double regular $n$-gon.

This contrasts with the conjectured existence of such points for the double regular heptagon. The source presents the claim as unresolved and notes that additional obstructions to periodic directions appear for $n=9$ and experimentally for larger odd $n$.