Positivity conjecture for Jack shifted functions and generalized Jack characters

Let $\lambda=\mathbf{r}^{\mathbf{p}}$ be a Young diagram in multirectangular coordinates. For a partition $\mu$, let $J^{\star,(\alpha)}_\mu$ be the shifted Jack function and let $\operatorname{\mathfrak{K}}^{(\alpha)}_\mu$ be the generalized Jack character defined from the monomial expansion of Jack polynomials. The $\alpha$-falling-factorial positivity conjecture concerns the basis

$$\left(\alpha^c(p_1)_{a_1}\cdots(p_d)_{a_d}(r_1)_{b_1}\cdots(r_d)_{b_d}\right)_{c,a_1,\dots,a_d,b_1,\dots,b_d\geq 0}$$

of the polynomial ring in $\alpha,p_1,\dots,p_d,r_1,\dots,r_d$. For every partition $\mu$, the quantities $\alpha^{|\mu|-\mu_1}J^{\star,(\alpha)}_\mu(\mathbf{r}^{\mathbf{p}})$ and $\operatorname{\mathfrak{K}}^{(\alpha)}_\mu(\mathbf{r}^{\mathbf{p}})$ are polynomials with non-negative rational coefficients in this $\alpha$-falling-factorial basis. The supplied source formulates this as a conjecture for general $\alpha$, and gives no resolution.