Dimension conjecture for discrete holomorphic quadratic differentials

Let $Q^{\mathbb{C}}(\operatorname{cr})$ and $Q^{\mathbb{R}}(\operatorname{cr})$ denote the complex and real spaces of discrete holomorphic quadratic differentials associated with a cross ratio system, and let $V$ be the vertex set of a closed triangulated surface of genus $g$. For a Delaunay cross ratio system $\operatorname{cr}$, the conjectured dimensions are

$$\dim_{\mathbb{C}} Q^{\mathbb{C}}(\operatorname{cr}) = \begin{cases} |V|+1 & \text{if } g=1,\\ |V|+6g-6 & \text{if } g\geq 2, \end{cases}$$

and

$$\dim_{\mathbb{R}} Q^{\mathbb{R}}(\operatorname{cr})=6g-6 \quad \text{if } g\geq 2.$$

This would determine the dimensions of the infinitesimal deformation spaces for Delaunay cross ratio systems beyond the already discussed sphere and torus cases; the supplied text gives no resolution of the conjecture.