Numerical zero-location conjecture for the dominant fusion QTM eigenvalues

Let $p_0$ be the parameter of the root-of-unity XXZ model, let $u$ be small, and let $T^{(1)}_{n-1}(u,v)$ denote the eigenvalue of the auxiliary quantum transfer matrix corresponding to the dominant eigenvector. The variable $v$ is complex, and $\text{mod }2p_0$ refers to the periodic identification of the imaginary part.

Zero-location conjecture. All the zeros of $T^{(1)}_{n-1}(u,v)$ are located on the almost straight lines

$$\operatorname{Im} v=\pm n\ \text{mod }2p_0.$$

This is a numerical analyticity assumption used to derive the integral equations for the free energy; the supplied text gives numerical motivation but no proof or resolution.