The symmetric-function conjecture for b-deformed Jucys–Murphy operators

Let $k$ be a positive integer, let $\mathcal{Z}\mathcal{V}_k$ be the subspace of vectors constant on coset-type, and let $\mathfrak{p}_\lambda,\mathfrak{w}_\lambda$ be the $b$-deformed class and idempotent analogues for partitions $\lambda\vdash k$. Let $\mathcal{J}_1,\ldots,\mathcal{J}_k$ be the $b$-deformed Jucys–Murphy operators, and let $\operatorname{cont}_b(\lambda)$ be the multiset of $b$-contents. Symmetric-function conjecture. For every symmetric function $f$ in $k$ variables and every partition $\lambda\vdash k$,

$$f(\mathcal{J}_1,\ldots,\mathcal{J}_k)\cdot\mathfrak{w}_\lambda=f(\operatorname{cont}_b(\lambda))\mathfrak{w}_\lambda.$$

Moreover, for each $\lambda$, $\mathfrak{w}_\lambda=\sum_{\mathsf{T}\in\mathsf{Tab}(\lambda)}\mathfrak{w}_{\mathsf{T}}$, and consequently every symmetric function of the operators preserves $\mathcal{Z}\mathcal{V}_k$. The conjecture is motivated by the corresponding symmetric-group and Jack-function identities; its validity remains open.