The representation-number conjecture for larger toroidal grid graphs

Let $\text{TGr}_{m,n}$ be the toroidal grid graph formed from the Cartesian product of the cycle graphs $C_m$ and $C_n$, with $m,n\geq 3$, and let $\mathcal{R}(G)$ denote the representation number of a graph $G$. Representation-number conjecture. If $n,m\geq 3$ and $m+n\geq 8$, then

$$\mathcal{R}(\text{TGr}_{m,n})\geq 4.$$

The graphs $\text{TGr}_{3,3}$ and $\text{TGr}_{3,4}$ have known 3-representations, but no 3-representation is known for larger toroidal grid graphs; the conjecture predicts that all cases with $m+n\geq 8$ require representation number at least $4$.