The unique minimal-$b$-invariant term conjecture for canonical basis vectors

Fix a multicharge $\mathbf{s}=(s_1,\ldots,s_l)\in\mathbb{Z}^l$ and let $\lambda$ be a cylindric $l$-partition. Expand the canonical basis vector $G(\lambda,\mathbf{s})$ in the standard basis as

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

where $a_{\lambda,\mu}(q)\in\mathbb{Z}[q,q^{-1}]$. Minimal-$b$-invariant conjecture. There exists a unique $\nu$ with minimal $b$-invariant among the $\mu$ for which $a_{\lambda,\mu}(q)\neq 0$. This conjecture proposes for canonical basis vectors an analogue of the unique minimal-$b$-invariant constituent known for Calogero–Moser cellular characters.