Non-periodic connection points on the double regular heptagon

A connection point is a point on a translation surface joined to a singularity by a saddle connection; a point is non-periodic if it is not periodic under the affine automorphism group. The double regular $n$-gon is the translation surface obtained by gluing two regular $n$-gons, with $n$ odd and $n\geq 7$.

Heptagon connection-point conjecture. There exist non-periodic connection points on the double regular heptagon.

The conjecture asks whether the double regular heptagon has connection points beyond the periodic points already understood; the source gives no resolution.