A positivity conjecture on the structure constants of shifted Jack functions

We consider Jack polynomials $J_\lambda$ and their shifted analogue J^#_\lambda. In 1989, Stanley conjectured that $\langle J_\mu J_\nu, J_\lambda \rangle$ is a polynomial with nonnegative coefficients in the parameter $\alpha$. In this note, we extend this conjecture to the case of shifted Jack polynomials.