Zero distribution conjecture for the second-largest spinless-fermion eigenvalue

Let $p_0\in 2\mathbb{Z}$, and let $T^{(2)}_{n-1}(v)$ denote the corresponding fused transfer-matrix eigenvalue for the second-largest eigenvalue in the sector $N_{\rm e}=N/2-1$. The physical strip is $\operatorname{Im}v\in[-1,1]$.

Zero distribution conjecture. $T^{(2)}_{n-1}(v)$ has one real zero $\zeta_{n-1}$ for $n$ even with $2\leq n\leq p_0-1$, and two real zeros $\zeta_{n-1}$ and $-\zeta'_{n-1}$ for $n$ odd with $2<n<p_0/2$. All other zeros are outside the physical strip.

This conjecture describes the analyticity pattern needed to derive the nonlinear integral equations for the second-largest eigenvalue. It is based on numerical studies, and the supplied passage gives no resolution beyond that numerical evidence.