Dimension conjecture for constant-bond-dimension-two tensor network varieties

Let $d\geq 3$, let $\mathbf{m}=(2,\dots,2)$, and let $\mathbf{n}=(n_1,\dots,n_d)$ with $n_j\geq 2$. For the tensor network variety $\mathcal{T\!N\!S}^{C_d}_{\mathbf{m},\mathbf{n}}$, one expects

$$\dim \mathcal{T\!N\!S}^{C_d}_{\mathbf{m},\mathbf{n}}=\min\left\{4\left(\sum_{j=1}^d n_j-d\right)+1,\prod_{j=1}^d n_j\right\}$$

except in the following cases: if $d=3$, $\mathbf{n}=(2,n_2,n_3)$ with $n_2\geq 3$, $n_3\geq 4$, and their cyclic permutations; or if $d=4$, $\mathbf{n}=(2,n_2,2,n_4)$ with $n_2,n_4\geq 2$, and their cyclic permutations.

Dimension conjecture. For all such $d$, $\mathbf{m}$, and $\mathbf{n}$, the displayed dimension formula holds outside precisely the listed exceptional cases.

The conjecture concerns the dimension of tensor network varieties for cycle graphs with constant bond dimension $2$. The authors report differential calculations at random points for $d=5,6,7$ supporting the formula, while the exceptional cases for $d=3,4$ are explicitly excluded; the general assertion remains open.