Uniqueness conjecture for the square grid $Z^c$

Let $Z^c$ be the square grid circle pattern with $0<c<2$, whose circles are labeled by

$$\mathbb V=\{z=N+iM:N,M\in\mathbb Z^2,\ M\geq |N|\},$$

and which covers the infinite sector with angle $c\pi/2$, with the centers of the border circles $C(N+iN)$ and $C(-N+iN)$ lying on the borders of that sector. Uniqueness conjecture. Up to rescaling, there is a unique square grid $Z^c$. The conjecture proposes a purely geometrical characterization of the discrete map $Z^c$, replacing its definition through an isomonodromy constraint; the source does not indicate whether the claim has been proved or disproved.