The $Q$-polynomial triple-intersection vanishing conjecture

Consider the $Q$-polynomial distance-regular graph $\Gamma=(X,\mathcal R)$, with standard module $V$, diameter $D$, and Krein parameters $q^h_{i,j}$. Let $\Lambda$ be the fundamental $\mathbb T$-submodule of $V^{\otimes 3}$, and let $E_h^{(1)},E_i^{(2)},E_j^{(3)}$ denote the corresponding tensor-factor operators. The $Q$-polynomial triple-intersection vanishing conjecture. For $0\leq h,i,j\leq D$,

$$E_h^{(1)} E_i^{(2)} E_j^{(3)} \Lambda=0 \qquad\text{if and only if}\qquad q^h_{i,j}=0.$$

This is the conjectural analogue for the ordinary primitive idempotents of the established statement involving the dual primitive idempotents and the intersection numbers $p^h_{i,j}$. It predicts that the vanishing pattern of these operator images on $\Lambda$ exactly records the vanishing of the Krein parameters.