Regular genus formula for the n-dimensional torus

Let $n$ be an integer with $n\geq 5$, and let $\Gamma^\prime$ be the $(n+1)$-regular colored graph constructed from the colored triangulation described in the paper. It has $(n+1)!$ vertices and represents the $n$-dimensional torus $\mathbb{S}^1 \times \mathbb{S}^1 \times \cdots \times \mathbb{S}^1$ ($n$ times). The regular genus conjecture. The graph $\Gamma^\prime$ is a genus-minimal crystallization of the $n$-dimensional torus, and its regular genus is

$$1+\frac{(n+1)!(n-3)}{8}.$$

The construction gives the asserted value for the displayed examples with $n\leq 4$, while genus-minimality for all $n\geq 5$ remains conjectural.