Pseudo-periodicity conjecture for the inhomogeneous eight-vertex ground-state eigenvector

Let $L=2n+1$, let $T_L(u|x_1,\ldots,x_L)$ be the inhomogeneous eight-vertex transfer matrix, and let $\Psi_L(x_1,\ldots,x_L)$ satisfy the eigenvector equations with eigenvalue $t_L(u|x_1,\ldots,x_L)$ and $F_\ast\Psi_L=(-1)^n\Psi_L$. Write $p=e^{\mathrm{i}\pi\tau}$ and $z_i=e^{-2\mathrm{i}x_i}$. Pseudo-periodicity conjecture. The eigenvector equations possess a solution $\Psi_L(x_1,\ldots,x_L)$ whose entries are theta functions of degree $L-1=2n$ and nome $p^2$ in each variable $x_i$, generically nonzero and without a common factor. Equivalently, the entries are holomorphic and satisfy

$$\Psi_L(\ldots,x_i+2\pi\tau,\ldots)=p^{-4n}z_i^{2n}\prod_{j\ne i}z_j^{-1}\Psi_L(\ldots,x_i,\ldots),$$ $$\Psi_L(\ldots,x_i+\pi,\ldots)=\prod_{j\ne i}\sigma_j\\,\Psi_L(\ldots,x_i,\ldots),$$

for $i=1,\ldots,L$, where the omitted arguments are the other variables. This conjecture describes the analytic structure of the inhomogeneous ground-state eigenvector; it is based on computations for $L=3,5,7$, and the source gives no resolution.