Sartori–Stroppel's conjecture on supersymmetric generators of the walled Brauer center

Let $B_{r,s}(\delta)$ be the walled Brauer algebra over $\mathbb C$, let $L_1,\ldots,L_{r+s}$ be its Jucys–Murphy elements, and let a supersymmetric polynomial mean a polynomial in the variables associated with the two sides of the wall that is symmetric separately in each set and satisfies the cancellation property. The supersymmetric polynomials in the $L_i$ are evaluated in these Jucys–Murphy elements.

Sartori–Stroppel's conjecture. For every $\delta\in\mathbb C$, the center of $B_{r,s}(\delta)$ is generated by the supersymmetric polynomials in $L_1,\ldots,L_{r+s}$.

The paper proves this assertion when the walled Brauer algebra is semisimple and presents the statement as the expected extension to all parameters. It is the specialization to the walled Brauer algebra of Sartori and Stroppel's conjecture for the walled Brauer category.