Modulo-two orbit conjecture for the Hecke group of the double heptagon

Let $\lambda_7=2\cos(\pi/7)$, let $H_7$ be the Hecke group acting on $\mathbb{Q}[\lambda_7]\cup\{\infty\}$, and let $\overline{H}_7$ be its reduction modulo two acting on $\mathbb{P}^1(\overline{\mathcal{O}})$. For $s=x/y\in\mathbb{Q}[\lambda_7]$, write $[\overline{x}:\overline{y}]$ for its reduced projective coordinates, and let $\Psi$ denote reduction modulo two.

Modulo-two orbit conjecture. An element $s=\frac{x}{y}\in\mathbb{Q}[\lambda_7]$ belongs to $H_7\cdot\infty$ if and only if $[\overline{x}:\overline{y}]$ belongs to the orbit of $[\overline{1}:\overline{0}]$ under $\overline{H}_7$. Equivalently,

$$H_7\cdot\infty=\Psi^{-1}(\overline{H}_7\cdot\Psi(\infty)).$$

If true, this would characterize connection points on the double heptagon through the parity of the denominator and the reduced Hecke-group orbit. The source gives no resolution.