Even-dimensional single-valuedness conjecture for graphical functions

Let $G$ be a weighted graph whose graphical function $f_G(z)$ exists in dimension $D=2\lambda+2\geq3$. Let $\mathcal{S}\mathcal{V}_{\{0,1,\infty\}}$ denote the space of functions on $\mathbb{C}\setminus\{0,1\}$ that are analytic in $z$ and $\overline{z}$ and have single-valued log-Laurent expansions at $0$, $1$, and $\infty$. The even-dimensional single-valuedness conjecture. Statement (G3) of Theorem 1 remains valid if the condition $D=4$ and $\nu_e=1$ for all $e\in\mathcal{E}_G$ is weakened to

$$D=2\lambda+2\geq4\text{ is even}\qquad\text{and}\qquad\lambda\nu_e\in\mathbb{Z}\text{ for all }e\in\mathcal{E}_G.$$

The claim extends the known single-valued log-Laurent expansion property from four-dimensional unit-weight graphs to even dimensions with integral rescaled edge weights. The supplied text gives no resolution, so the conjecture remains open.