Cross-ratio rigidity conjecture for surfaces in \mathcal{S}

Let $\mathcal{R}$ and $\mathcal{R}'$ be two surfaces in $\mathcal{S}$. Suppose the cell of $\mathcal{R}$ at position $(i,j)$ has cross ratio $k_{i,j}$, while the cell of $\mathcal{R}'$ at position $(i,j)$ has cross ratio $k'_{i,j}$. Assume that $k_{i,j}=k'_{i,j}$ for all $i,j$. Cross-ratio conjecture. Then $\mathcal{R}$ and $\mathcal{R}'$ have the same conformal type; equivalently, they are equivalent up to biholomorphism. This would show that the cross ratios determine the conformal type of a surface and would facilitate constructing probability distributions on the full space of surfaces with a square grid net, potentially advancing the study of parabolicity.