Irreducibility criterion for the associated quantum affine module

Let $U_q(\widehat{\mathfrak{sl}}_2)$ be the quantum affine algebra and let $V$ be the standard module constructed from the tridiagonal-pair data, with weights $w_1,\ldots,w_d$. Quantum-module irreducibility conjecture. The module $V$ is irreducible if and only if

$$w_i\ne q^2w_j\qquad(1\leq i,j\leq d).$$

The claim is stated as a conjecture in the source, and the supplied text gives no proof or resolution.