Lexicographic quasi-idempotent Schur duality for polynomial quantum wreath products

Assume that $B\wr \cH_d$ is a polynomial quantum wreath product satisfying conditions

$$and$$

, and write $\bfH_d=B\wr \cH_d$. For each composition $\lambda\vDash d$, let $K_\lambda$ be defined by

with the product taken in lexicographic order. Define the \right $\bfH_d$-module

\bfC^\cT=\bigoplus_{\lambda}K_\lambda\bfH_d.

$$Let$$

\overline{\bfS}^\cT_d=\bigoplus_{\lambda,\mu}\overline{\bfS}^\cT_{\lambda,\mu},\qquad \overline{\bfS}^\cT_{\lambda,\mu}=K_\lambda\bfH_d\cap\bfH_dK_\mu,

with multiplication $(xK_\mu)\cdot(K_\mu y)=xK_\mu y$ for $xK_\mu\in\overline{\bfS}^\cT_{\lambda,\mu}$ and $K_\mu y\in\overline{\bfS}^\cT_{\mu,\nu}$. **Lexicographic quasi-idempotent Schur duality conjecture.** The statement of Theorem

holds for this construction. This proposes that the strong double-centralizer property extends when only conditions

$$and$$

are imposed, despite the failure of the convolution-algebra approach without centrality. The source gives no resolution of this conjecture.