The b-conjecture for Jack map coefficients

Let $J^{(\alpha)}_\lambda$ be Jack polynomials and let $c^\pi_{\mu,\nu}(\alpha)$ be defined by

$$\sum_{\lambda\in\mathbb{Y}}u^{|\lambda|}\frac{J^{(\alpha)}_\lambda[X]J^{(\alpha)}_\lambda[Y]J^{(\alpha)}_\lambda[Z]}{j^{(\alpha)}_\lambda}=\sum_{m\geq0}\sum_{\pi,\mu,\nu\vdash m}\frac{u^mc^\pi_{\mu,\nu}(\alpha)}{z_\pi\alpha^{\ell(\pi)}}p_\pi[X]p_\mu[Y]p_\nu[Z].$$

Set $b:=\alpha-1$.

The b-conjecture. The coefficients $c^\pi_{\mu,\nu}$ are polynomials in $b$ with non-negative integer coefficients. The paper states that the conjecture remains open, although its integrality part has been proved.