Non-degeneracy conjecture for the simple nineteen-vertex-model eigenvalue

Let $T^{(2)}(z)$ be the transfer matrix of the nineteen-vertex model with twist angle $\phi=\pi$, and let

$$\theta^{(2)}(z|w_1,\dots,w_N)=(-1)^{N+1}\prod_{j=1}^N [q w_j/z][q^2 z/w_j]$$

be its eigenvalue for inhomogeneity parameters $w_1,\dots,w_N$. Non-degeneracy conjecture. Except for possibly a finite number of values of the parameter $q$, this eigenvalue is non-degenerate for all $N$. Numerical diagonalisation for small system sizes suggests non-degeneracy, but a general proof is not provided.