Form-factor formulas for partial zero modes in GL(N)-invariant models

Let $T^{(1)}_{ij}[0]$ be the partial zero modes in a ${\rm GL}(N)$-invariant model. Bethe vectors and dual Bethe vectors are parametrized by sets $\bar t^k$ and cardinalities $a_k$, and let $\alpha_k(t)=\lambda_k^{(1)}(t)/\lambda_{k+1}^{(1)}(t)$ for $k=1,\dots,N-1$. For a set of $\kappa$-twisted on-shell Bethe parameters $\bar t(\bar\kappa)$, assume $\bar t(\bar\kappa)=\bar t$ when $\bar\kappa=1$. The universal form factor of the total operator $T_{ij}(z)$ is denoted by ${\mathfrak{F}}_{\bar a}^{(i,j)}(\bar s;\bar t)$. Form-factor conjecture. Form factors of the partial zero modes are given by

$$\mathbb{C}_{\bar b}(\bar s) T^{(1)}_{ij}[0] \mathbb{B}_{\bar a}(\bar t) = \left(\prod_{k=1}^{N-1}\frac{\alpha_k(\bar s^{k})}{\alpha_k(\bar t^{k})}-1\right){\mathfrak{F}}_{\bar a}^{(i,j)}(\bar s;\bar t), \qquad \text{for}\quad \bar s \neq\bar t$$

and

$$\mathbb{C}_{\bar a}(\bar t) T^{(1)}_{ii}[0] \mathbb{B}_{\bar a}(\bar t) = \left( \lambda^{(1)}_{i}[0]+\sum_{k=1}^{N-1}\frac{d}{d\kappa_i} \log\alpha_k\big(\bar t^{k}(\bar\kappa)\big) \Bigr|_{\bar\kappa=1}\right)\|\mathbb{B}_{\bar a}(\bar t)\|^2.$$

These formulas would extend the form-factor results from ${\rm GL}(3)$-invariant models to ${\rm GL}(N)$ models. The candidate is presented without evidence of resolution, so its status remains open.