Analytic properties of the density matrix for the $\mathfrak{sl}_{n+1}$-invariant model

Let $D_{1,\dots,m}(\lambda_1,\dots,\lambda_m)$ be the density matrix, with spectral parameters $\lambda_1,\dots,\lambda_m\in\mathbb{C}$. For $0<\delta<\pi$, define

$$S_\delta:=\{\lambda\in\mathbb{C}\mid \delta<|\arg(\lambda)|<\pi-\delta\}.$$

Analytic-properties conjecture. The function $D_{1,\dots,m}$ is meromorphic in $\lambda_1,\dots,\lambda_m$, with at most simple poles when

$$\lambda_i-\lambda_j\in\mathbb{Z}\backslash\{0,\pm1,\dots,\pm n\},$$

and satisfies

$$\lim_{\substack{\lambda_1\to\infty\lambda_1\in S_\delta}}D_{1,\dots,m}(\lambda_1,\dots,\lambda_m)=\frac{1}{n+1}\boldsymbol{1}_1D_{2,\dots,m}(\lambda_2,\dots,\lambda_m).$$

These properties are expected to follow from integral formulas obtained by the vertex-operator approach in the massive regime and the limit $q\to1$. They have been checked for $\mathfrak{sl}_3$ for the one-, two- and three-point density matrices, but a complete proof is left open.