Lassalle's positivity conjecture for Jack character polynomials

Let $\lambda$ be a partition and let $\theta^{(\alpha)}_\mu(\lambda)$ be the Jack-character coefficient defined by the paper, with $b:=\alpha-1$. Let $m_1(\mu)$ denote the number of parts of $\mu$ equal to $1$, and let $(q_1,q_2,\ldots,r_1,r_2,\ldots)$ be the multirectangular coordinates of $\lambda$. Lassalle's positivity conjecture. For every partition $\mu$ of size $m$ with $m_1(\mu)=0$, the quantity $(-1)^{|\mu|}z_\mu\theta^{(\alpha)}_\mu(\lambda)$ is a polynomial in $(b,q_1,q_2,\ldots,-r_1,-r_2,\ldots)$ with non-negative integer coefficients. The source states that polynomiality in $\alpha$ and in the multirectangular coordinates is known, while this positivity property remains a conjecture.