Strength bound for dual-thin Q-polynomial association schemes

Prove that, for every dual-thin Q-polynomial association scheme with at least three classes and a corresponding Q-polynomial idempotent of multiplicity at least three, the associated spherical embedding has strength at most seven, and show that strength seven forces the embedding to be a tight spherical 7-design and the association scheme to be Q-bipartite with four classes.

Background

The conjecture restricts the preceding uniform-bound problem to the subclass of dual-thin Q-polynomial association schemes, whose Terwilliger algebras have all irreducible modules dual thin. This condition is known to impose strong structural restrictions, analogous to thinness for distance-regular graphs.

The proposed bound is seven, with an asserted extremal characterization: equality should imply that the spherical embedding is a tight spherical 7-design, equivalently in the stated setting that the scheme is Q-bipartite with diameter (number of classes) four. The paper notes that dual-thin Q-polynomial schemes with a_1*=0 are already known to be Q-bipartite or almost Q-bipartite, but the full conjecture remains unresolved.

References

it seems safe to make the following conjecture: Referring to Assumption~\ref{assumption: Q only}, assume that $(X,\mathcal{R})$ is dual thin, $d\geqslant 3$, and $m\geqslant 3$. Then, the strength $t(\tilde{X})$ of $\tilde{X}$ satisfies $t(\tilde{X})\leqslant 7$. Moreover, if $t(\tilde{X})=7$, then $\tilde{X}$ is a tight spherical $7$-design (so $(X,\mathcal{R})$ is $Q$-bipartite with $d=4$ by Proposition~\ref{antipodal iff Q-bipartite}).

— On the spherical design properties of a $P$- and $Q$-polynomial association scheme  (2608.27082 - Lansdown et al., 27 Aug 2026) in Section 5, Concluding remarks; Conjecture 5.5