Connection-point characterization for the double heptagon

Let $\lambda_7=2\cos(\pi/7)$, and consider the associated staircase model of the double heptagon. Let $P$ have coordinates $\frac{1}{N}(x,y)$, where $N\in\mathbb{N}^*$ and $x,y\in\mathbb{Z}[\lambda_7]$ are not both divisible by a common divisor of $N$. Let $\overline{H}_7$ be the reduced Hecke group acting on projective reductions modulo two.

Double-heptagon connection-point conjecture. The point $P$ is a connection point if and only if $N$ is even and $[\overline{x}:\overline{y}]$ belongs to the orbit of $[\overline{1}:\overline{0}]$ under $\overline{H}_7$.

This is presented as a more precise form of the conjectured existence of non-periodic connection points on the double heptagon, conditional in the surrounding discussion on the modulo-two orbit conjecture. The source gives no resolution.