The q-tetrahedron algebra realization conjecture for tridiagonal pairs

Assume $\mathbb K$ is algebraically closed. Let $V$ be a finite-dimensional vector space over $\mathbb K$ of positive dimension, and let $A,A^*$ be a tridiagonal pair on $V$. Let $\beta=q^2+q^{-2}$, where $\beta$ is the scalar from the tridiagonal-pair relations, and assume that $q$ is not a root of unity. Let $\boxtimes_q$ be the $q$-tetrahedron algebra. The q-tetrahedron algebra realization conjecture. There exists an irreducible $\boxtimes_q$-module structure on $V$ such that $A$ acts as a linear combination of $x_{01},x_{12},I$, and $A^*$ acts as a linear combination of $x_{23},x_{30},I$. This would extend the established realization for tridiagonal pairs of $q$-geometric type to the stated general setting. The supplied source gives no resolution.