Universal K-matrix specialization conjecture for the q-Onsager algebra

Let ${\bf K}^{(\frac{1}{2})}(u)$ and ${\cal K}^{(\frac{1}{2})}(u)$ be, respectively, the spin-$\frac{1}{2}$ specialization of a universal K-matrix and the fused K-operator, with ${\cal K}^{(\frac{1}{2})}(u)$ defined in~. Let $\nu(u)$ be an invertible central element of ${\cal A}_q[[u^{-1}]]$, and let $\pi^{\frac{1}{2}}(\mu(u))$ and $\Gamma(u)$ be the functions defined in~ and~. Universal K-matrix specialization conjecture. One has

$${\bf K}^{(\frac{1}{2})}(u)=\nu^{(\frac{1}{2})}(u){\cal K}^{(\frac{1}{2})}(u),$$

where $\nu(u)$ satisfies

$$\pi^{\frac{1}{2}}(\mu(u^2q))\nu(u)\nu(uq)\Gamma(u)=1.$$

This conjecture is the spin-$\frac{1}{2}$ case of the universal K-matrix specialization proposed in~; the corresponding statement for higher spins is described there as a corollary. It is used to derive universal TT-relations, but the supplied text gives no resolution of the conjecture.