Let $P$ be a fixed pentagon in the hyperbolic plane $\mathbb H^2$ with all the angles equal to $\pi / 3$. Let $w_1, w_2, \dots, w_5$ be the sides of $P$ going in the counterclockwise order. We are looking for five holomorphic functions $f_1, f_2, \dots f_5 \colon \mathbb H^2 \to \mathbb C$ such that $\sum_{j=1}^5 f_j(z) \equiv 1$ and if $z \in w_j$ for some $j$ then the following numbers are real

$$f_j(z),\quad \tau f_{j+1} (z),\quad \overline \tau f_{j-1} (z),\quad f_{j+1}(z) + f_{j-2}(z), \quad f_{j-1}(z) + f_{j+2}(z) \in \mathbb R,$$ where $\tau = \exp(2\pi i / 3)$. Is it true that the space of solutions $(f_1, f_2, f_3, f_4, f_5)$ has finite dimension? Is it true that there is at most one solution?