Tautness conjecture for pseudo primitive idempotents

Let $\Gamma$ be a bipartite distance-regular graph, and let $T$ denote its subconstituent algebra. Assume that, up to isomorphism, there exist exactly two irreducible $T$-modules with endpoint $2$, and that both are thin. Choose $\xi,\chi\in\mathbb C\cup\{\infty\}$ such that $\widetilde{\xi}$ and $\widetilde{\chi}$ are the local eigenvalues of these modules. Let $E$ and $F$ be pseudo primitive idempotents of $\Gamma$ for $\xi$ and $\chi$, respectively. Tautness conjecture. The pair $E,F$ is taut, meaning that $E\circ F$ is a linear combination of at most two pseudo primitive idempotents of $\Gamma$. This conjecture predicts a connection between the thin endpoint-$2$ modules of the subconstituent algebra and taut pairs of pseudo primitive idempotents; its resolution is not specified in the source.