Closedness conjecture for uniform matrix product state varieties

Let $D$, $d$, and $N$ be positive integers. Write $\operatorname{uMPS}(D,d,N)$ for the variety of uniform matrix product states, and let

$$\operatorname{Cyc}^N(\mathbb{C}^d)$$

be the ambient cyclic tensor space. The trivial cases are those listed in Proposition.

Closedness conjecture. Except for the trivial cases in Proposition, $\operatorname{uMPS}(D,d,N)$ is only closed if it fills the ambient space $\operatorname{Cyc}^N(\mathbb{C}^d)$.

The conjecture predicts that, apart from explicitly identified trivial cases, a proper uniform matrix product state variety is not closed. The paper proves non-closedness in broad ranges using the injectivity radius, but the full assertion is not resolved there.