The meromorphicity conjecture for spectral K-matrices

Let $\mathcal{C}$ be the category of finite-dimensional modules under consideration, let $V\in\mathcal{C}$, and let $K_V(z)$ be the matrix-valued spectral universal K-matrix acting on $V$. Assume $q\in\mathbb{C}^{\times}$ satisfies $1\notin q^{\mathbb{Z}}$, and take generic quantum symmetric pair parameters. A meromorphic matrix-valued function is a matrix whose entries are meromorphic functions of $z$.

Meromorphicity conjecture. For all $V\in\mathcal{C}$, not necessarily irreducible, and for generic quantum symmetric pair parameters, $K_V(z)$ is the Laurent series expansion of a meromorphic matrix-valued function.

For irreducible modules, the intertwining condition has a one-dimensional solution space and yields trigonometric K-matrices. The conjecture asks for meromorphic dependence for arbitrary objects of the category; the source does not report a resolution.