The spetsial-parameter coefficient conjecture for canonical basis vectors

Let $\mathbf{s}=(s+1,s,\ldots,s)$ be the spetsial multicharge, and let $\lambda$ be a cylindric $l$-partition. Write

$$G(\lambda,\mathbf{s})=\sum_{\mu}a_{\lambda,\mu}(q)\ket{\mu,\mathbf{s}},$$

and let $m(\lambda)$ be the multipartition selected by the minimal-$b$-invariant conjecture. For each $l$-partition $\mu$, let $\xi_\mu$ be the leading coefficient of the Schur element associated with the corresponding representation of the Ariki–Koike algebra. Spetsial-parameter coefficient conjecture. One has

$$a_{\lambda,m(\lambda)}(1)=\sum_{\mu}\frac{a_{\lambda,\mu}(1)}{\xi_\mu}.$$

The assertion concerns the relation between canonical-basis coefficients and Schur-element data at spetsial parameters.