The Laplace–Beltrami correspondence for b-deformed Jucys–Murphy operators

Let $k$ be a positive integer, let $\mathcal{Z}\mathcal{V}_k$ be the coset-type-constant subspace, let $\mathfrak{p}_\lambda$ be the $b$-deformed class vector for $\lambda\vdash k$, and let $\operatorname{ch}^{(b)}$ be the $b$-deformed Frobenius characteristic map. Let $D(b)$ denote the Laplace–Beltrami operator. Laplace–Beltrami correspondence conjecture. For every partition $\lambda\vdash k$,

$$\operatorname{ch}^{(b)}\big((\mathcal{J}_1+\mathcal{J}_2+\cdots+\mathcal{J}_k)\cdot\mathfrak{p}_\lambda\big)=D(b)\cdot\operatorname{ch}^{(b)}(\mathfrak{p}_\lambda).$$

This conjecture seeks to identify the sum of the deformed Jucys–Murphy operators, after applying the characteristic map, with the Laplace–Beltrami operator whose eigenfunctions are Jack symmetric functions. It is motivated by the classical correspondence for symmetric groups and is unresolved.