Zero-location conjecture for fused transfer-matrix eigenvalues

Let $u$ be the parameter of the quantum transfer matrix, let $r\geq 1$, and let $a\in\{1,2,\dots,r\}$ and $m\in\mathbb{Z}_{\geq 1}$. Denote by $\widetilde{T}^{(a)}_{m}(v)$ the fused transfer-matrix eigenvalue and by $\{\tilde{z}^{(a)}_{m}\}$ its zeros. The chemical potentials are $\mu_1,\mu_2,\dots,\mu_{r+1}$. Zero-location conjecture. For small $u$ ($|u|\ll 1$), every zero $\{\tilde{z}^{(a)}_{m}\}$ of $\widetilde{T}^{(a)}_{m}(v)$ is located near the lines

$$\operatorname{Im}v=\pm\frac{m+a}{2}$$

at least when $\mu_1=\mu_2=\dots=\mu_{r+1}=0$. This describes the zero pattern underlying the proposed $Y$-system analysis of the one-string solution; the source provides numerical evidence for the claim, but no proof or resolution is given.