The q-tetrahedron action conjecture for classical-parameter graphs

Let $\Gamma$ be a distance-regular graph with classical parameters $(D,b,\alpha,\beta)$, where $b\ne1$, and suppose $\Gamma$ is not a dual polar graph. Let $T$ be its Terwilliger algebra, $Z(T)$ its center, and let $x_{01},x_{12},x_{23}$ denote the indicated generators of the $q$-tetrahedron algebra. The $q$-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 $1,x_{23}$. This proposes a natural quantum-algebra action for a specified class of distance-regular graphs, beyond the cases already covered by the paper's assumptions.