The orbit-representation conjecture for integral points of $X_{N,y}$

Let $N\in\mathbb{N}$ and $y\in\Gamma_N$. Let $\eta\in X_{N,y}(\mathbb{Z})$ be an integral point. For $\widehat{w}\in\Sigma W^0$ and $g\in G$, define the action $g\cdot\widehat{w}$ as in the preceding notation. Orbit-representation conjecture. There exist $\widehat{w}\in\Sigma W^0$ and $g\in G$ such that

$$g\cdot\widehat{w}=\eta.$$

This conjecture is presented as a natural analogue of the earlier result referred to as the conjecture in the source, and has been checked for the first values of $N$ up to $N=1000$. Its general validity remains open.