Jack polynomial dominance conjecture

Let $\lambda$ and $\mu$ be partitions with $|\lambda|=|\mu|$, and let $P_\lambda(x;\tau)$ denote the Jack polynomial in $n$ variables. For $\tau_0\in[0,\infty]$, define

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

Jack polynomial dominance conjecture. The following statements are equivalent: $\lambda$ dominates $\mu$; for every $x\in[0,\infty)^n$,

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

and, for every $x\in[0,\infty)^n$,

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

This conjecture extends known dominance inequalities for classical symmetric polynomials to Jack polynomials; the supplied text does not state whether it has been proved or disproved.