Improper marginalization birationality conjecture for translation-invariant matrix product states

Let $N\geq 4$. An improper marginalization maps a translation-invariant $N$-qubit matrix product state variety $\overline{\operatorname{OB}}(2,2,N)$ to $\overline{\operatorname{OB}}(2,2,3)$ by summing over all but three consecutive indices. Improper marginalization birationality conjecture. A generic $N$-qubit state can be recovered from its improper marginalization to any three consecutive states; equivalently, each such map

$$\overline{\operatorname{OB}}(2,2,N)\to\overline{\operatorname{OB}}(2,2,3)$$

is a birational equivalence of varieties. The analogous statement for hidden Markov models is known, while this matrix product state version concerns whether the loss of information under improper marginalization is generically insignificant.