Zinn-Justin's eigenvector conjecture for the supersymmetric eight-vertex model

Let $L=2n+1$ with $n\geqslant 0$, let $\eta=\pi/3$, and let $V$ be the two-dimensional spin space. For inhomogeneity parameters $u_1,\dots,u_{2n+1}$, consider the eigenvalue problem

$$T(u|u_1,\dots,u_{2n+1})|\Psi\rangle=\Theta_n(u|u_1,\dots,u_{2n+1})|\Psi\rangle,\qquad F|\Psi\rangle=(-1)^n|\Psi\rangle,$$

with $|\Psi\rangle\in V^{2n+1}$. Here $p$ is the elliptic nome, the $\sigma_j^z$ are Pauli spin operators, and $\dots$ denotes omission of the indicated inhomogeneity. Zinn-Justin's conjecture. The solution space is spanned by a vector

$$|\Psi_n\rangle=|\Psi_n(u_1,\dots,u_{2n+1})\rangle\in V^{2n+1},$$

whose components are entire functions of each $u_i$, are generically non-vanishing, and have no common factor depending on $u_1,\dots,u_{2n+1}$. Moreover, for every $i=1,\dots,2n+1$,

$$|\Psi_n(\dots,u_i+2\pi\tau,\dots)\rangle=p^{-4n}e^{-2\mathrm{i}\sum_{j=1}^{2n+1}(u_i-u_j)}|\Psi_n(\dots,u_i,\dots)\rangle,$$ $$|\Psi_n(\dots,u_i+\pi,\dots)\rangle=\prod_{j=1,j\neq i}^{2n+1}\sigma_j^z|\Psi_n(\dots,u_i,\dots)\rangle.$$

The conjecture characterizes the inhomogeneous eigenvector associated with the doubly degenerate supersymmetric eigenvalue and asserts its analytic and quasiperiodic properties. Existence is known for small $n$, while a rigorous proof for arbitrary $n$ was not established in the source.