Conjecture A+ on Kazhdan–Lusztig cells for unequal parameters

Let $W_n$ be the Coxeter group of type $B_n$ with weight function $L_{a,b}$, where $a,b\in\Gamma_{>0}$, and let $P^r(w)$ and $Q^r(w)$ be the domino insertion and recording tableaux defined using the $2$-core $\delta_r$.

Conjecture A+. Let $r\geqslant 0$ and assume that $ra<b<(r+1)a$. Then the Kazhdan–Lusztig left, right, and two-sided cells for these parameters coincide with those obtained for the special values $a=2$ and $b=2r+1$ over $\Gamma=\mathbb{Z}$; equivalently, the statements of Conjecture A hold.

This extends the proposed domino-insertion description from the special integral parameters to every parameter pair in the open interval $ra<b<(r+1)a$. The claim is supported by computations and by known results in the asymptotic case, but remains conjectural in general.