Numerical zero-location conjecture for the third QTM eigenvalue

Let $p_0\in\mathbf Z_{\ge 3}$, let $u$ be small and negative, and let $T^{(3)}_{n-1}(u,v)$ denote the eigenvalue associated with the third-largest quantum transfer-matrix eigenvalue. For $2\le n\le p_0$, regard $T^{(3)}_{n-1}(u,v)$ as a function of complex $v$.

Zero-location conjecture. For $n<p_0$, $T^{(3)}_{n-1}(u,v)$ has two real zeros $\pm\zeta^{(3)}_{n-1}$ for some $\zeta^{(3)}_{n-1}\in\mathbf R_{>0}$, while for $n=p_0$ it has a double zero at $\zeta^{(3)}_{p_0-1}=0$. All the other zeros of $T^{(3)}_{n-1}(u,v)$, for $2\le n\le p_0$, are located on the almost straight lines

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

The claim is based on numerical studies and describes the zero pattern relevant to the third correlation-length eigenvalue; the supplied text gives no proof or resolution.