The q-tetrahedron action conjecture for type I Q-polynomial graphs

Let $\Gamma$ be a $Q$-polynomial distance-regular graph with eigenvalues $\theta_i$ and dual eigenvalues $\theta_i^*$. For $2\leq i\leq D-1$, define

$$\frac{\theta_{i-2}-\theta_{i+1}}{\theta_{i-1}-\theta_i},\qquad \frac{\theta^*_{i-2}-\theta^*_{i+1}}{\theta^*_{i-1}-\theta^*_i}.$$

Assume these expressions are equal and independent of $i$, with common value $b+b^{-1}+1$, assume $b$ is not a root of unity, and assume the given $Q$-polynomial structure is type I in the sense of Bannai and Ito, with $s\ne0$ and $s^*\ne0$. Let $T$ be the Terwilliger algebra, $Z(T)$ its center, and $x_{01},x_{12},x_{23},x_{30}$ the indicated $q$-tetrahedron generators. The type I $q$-tetrahedron action conjecture. If $b=q^2$, then there exists a $boxtimes_q$-action on the standard module of $\Gamma$ such that the adjacency matrix is a $Z(T)$-linear combination of $1,x_{01},x_{12}$, while the dual adjacency matrix is a $Z(T)$-linear combination of $1,x_{23},x_{30}$. This predicts a quantum-algebra action under the stated type I spectral conditions.