The minimal-content conjecture for canonical basis vectors in Calogero-Moser families

Let $\mathbf{s}$ be a multicharge of the form

$$\mathbf{s}=(s+1,\ldots,s+1,s,\ldots,s).$$

Let $\lambda$ be a cylindric $l$-partition, and let $m(\lambda)$ denote the unique $l$-partition with minimal $b$-invariant occurring with nonzero coefficient in the standard-basis expansion of $G(\lambda,\mathbf{s})$. Minimal-content conjecture. The multipartition $m(\lambda)$ is the symbol of minimal content in the Calogero–Moser family containing $\lambda$. This predicts that the minimal-$b$-invariant constituent is determined by the corresponding family of symbols for these parameters.