Connection points on double regular polygons

We study connection points on the double regular $n$-gon translation surface, for $n\ge 7$ odd and its staircase model. For $n\mathrm{\ne }9$, we provide a large family of points with coordinates in the trace field that are not connection points. This family includes the central points, and for $n=7$ we conjecture that all the remaining points are connection points. Further, in the case where $n\ge 7$ is a prime number, we provide a constructive proof by exhibiting an explicit separatrix passing through a central point that does not extend to a saddle connection.