Tridiagonal-pair and type-(1,1) quantum-affine-module correspondence

Let $A,A^*$ be a tridiagonal pair on $V$ satisfying the conditions of Definition 1p1 and the stated eigenvalue-form lemma. Let $K,R,L$ denote the operators in the associated $U_q(\widehat{\mathfrak{sl}}_2)$ action, and call a finite-dimensional irreducible module type $(1,1)$ when $K_0K_1$ acts as the identity. Type-(1,1) correspondence conjecture. The space $V$ has an irreducible type-$(1,1)$ $U_q(\widehat{\mathfrak{sl}}_2)$-module structure such that

$$A=aK+R,\qquad A^*=a^*K^{-1}+L,$$

and none of its weights equals $aa^*(q-q^{-1})^2$. Conversely, if $V$ is a finite-dimensional irreducible type-$(1,1)$ module, $a,a^*$ are nonzero scalars, and none of its weights equals $aa^*(q-q^{-1})^2$, then the displayed formulas define a tridiagonal pair satisfying the same conditions. The conjecture proposes a classification correspondence between the specified tridiagonal pairs and quantum-affine modules; the supplied text gives no resolution.