Conjecture on the defining ideal of orthogonal tensor trains

Let $SOT_{2,n}$ denote the set of orthogonal tensor trains of length $2$ in $\mathbb{C}^{n^4}$, and let $\mathcal P_n$, $\mathcal Q_n$, and $h_n$ be the polynomial families and polynomial defined in the preceding results. The Zariski closure of $SOT_{2,n}$ is the algebraic set defined by the vanishing of these polynomials.

Defining-ideal conjecture. The Zariski closure of $SOT_{2,n}$ in $\mathbb{C}^{n^4}$ is cut out by the vanishing of $\mathcal P_n$, $\mathcal Q_n$, and $h_n$. The ideal defined by these equations is prime.

This conjecture proposes a complete set-theoretic and scheme-theoretic description of the Zariski closure of $SOT_{2,n}$; the preceding discussion gives computational evidence for $n=2$ and $n=3$, while the general case remains open.