The zero-location conjecture for fused transfer-matrix functions

Let $u$ be sufficiently small, with $|u|\ll 1$, let $s$ be a positive integer, and let $a\in\{1,2,\dots,s\}$. For each positive integer $m$, let $T^{(a)}_{m}(v)$ denote the transfer-matrix function defined in the preceding construction, and call

$$\{v\in\mathbb{C}:\operatorname{Im}v\in[-\tfrac{1}{2},\tfrac{1}{2}]\}$$

the physical strip. Zero-location conjecture. Every zero of $T^{(a)}_{m}(v)$ is located outside the physical strip.

This analyticity property is expected for the two-string solution that provides the largest quantum-transfer-matrix eigenvalue in the sector $N=M_{1}=M_{2}=\cdots=M_{s}$, based on numerical analysis for finite $N$, $u$, and $s$. The source gives no proof or resolution of the conjecture.