The b-conjecture for connected Jack map coefficients

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

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

Set $b:=\alpha-1$.

The b-conjecture. The coefficients $h^\pi_{\mu,\nu}$ are polynomials in $b$ with non-negative integer coefficients. The paper explicitly says both Jack conjectures remain open.