The Norton-balanced kite-function conjecture for Q-polynomial distance-regular graphs

Let $\Gamma=(X,\mathcal R)$ be a $Q$-polynomial distance-regular graph with diameter $D\geq 3$, and let $E$ be a $Q$-polynomial primitive idempotent of $\Gamma$. Assume that the set $\{E\hat{x}\mid x\in X\}$ is Norton-balanced. The Norton-balanced kite-function conjecture. The kite function $\zeta_i$ is constant for $2\leq i\leq D$. This conjecture concerns the structure of Norton-balanced $Q$-polynomial distance-regular graphs; the supplied text does not indicate whether it has been proved or disproved.