Parameter-inversion conjecture for universal K-matrices

Let $A$ be of affine type and let $(X,\tau)\in\mathcal{G}\operatorname{Sat}(A)$. Suppose $(Y,\eta)\in\mathcal{G}\operatorname{Sat}_\tau(A)$. Set $\psi=\psi^g_{Y,\eta}$ for $g\in(U_q\mathfrak h)^{\mathcal W,\times}$ and set $\phi=\eta\circ\tau$. For a finite-dimensional type-1 representation $V(z)$ of $U_q\mathfrak g'$, consider the parameter-inversion relation

$$(\pi^V_z\circ\psi)(u)=(\pi^V_{1/z}\circ\phi)(u)\qquad\text{for all }u\in U_q\mathfrak g'.$$

Parameter-inversion conjecture. There exists $g\in(U_q\mathfrak h)^{\mathcal W,\times}$ such that this relation is satisfied. Consequently, $k_{Y,\eta}$ specializes to a solution of the parameter-dependent reflection equation. The conjecture would identify the required twist automorphism for finite-dimensional type-1 representations and thereby produce spectral reflection-equation solutions from universal K-matrices; the source supplies no resolution.