Jack polynomial weak-dominance conjecture

Let $\lambda$ and $\mu$ be partitions, and let $P_\lambda(x;\tau)$ denote the Jack polynomial in $n$ variables. For $\tau_0\in[0,\infty]$, consider $x\in[1,\infty)^n$ and the coefficient ring $\mathbb F_{\geqslant0}^{\mathbb R}$ defined by

$$\mathbb F_{\geqslant0}^{\mathbb R}=\{f/g\mid f(\tau)\in\mathbb R_{\geqslant0}[\tau],\ g\in\mathbb Z_{\geqslant0}[\tau]\setminus\{0\}\}.$$

Jack polynomial weak-dominance conjecture. The following statements are equivalent: $\lambda$ weakly dominates $\mu$; for $x\in[1,\infty)^n$,

$$\frac{P_\lambda(x;\tau)}{P_\lambda(\bm1;\tau)}-\frac{P_\mu(x;\tau)}{P_\mu(\bm1;\tau)}\in\mathbb F_{\geqslant0}^{\mathbb R};$$

and, for $x\in[1,\infty)^n$,

$$\frac{P_\lambda(x;\tau_0)}{P_\lambda(\bm1;\tau_0)}-\frac{P_\mu(x;\tau_0)}{P_\mu(\bm1;\tau_0)}\geqslant0.$$

This is the Jack-polynomial analogue of the preceding weak-dominance inequalities for Schur and related symmetric polynomials; the supplied text gives no resolution status.