The factorized $B'_2$-Leonard pair conjecture for the quantum loop algebra construction

Let $V_{2j_1}(u)\tensor V_{2j_2}(u)$ be the tensor-product vector space, and let $(\mathcal{A}_2,\mathcal{A}_{12};\mathcal{B}_{12},\mathcal{B}_1)$ be the indicated ordered 4-tuple of operators. Assume that this vector space is irreducible under their action. Factorized $B'_2$-Leonard pair conjecture. Then $(\mathcal{A}_2,\mathcal{A}_{12};\mathcal{B}_{12},\mathcal{B}_1)$ is a basis of a factorized $B'_2$-Leonard pair. If true, this would give the coefficients $T_{k_1,k_2}(\ell_1,\ell_2)$ at $u_1=u_2$ a natural interpretation as overlap coefficients for such a pair.