KT conjecture for Jack polynomials

Let $\lambda$ and $\mu$ be partitions, and let $P_\lambda(x;\tau)$ denote the Jack polynomial. Write $\bm1=(1,\ldots,1)$. KT conjecture. The following are equivalent:

$$\frac{P_\lambda(x+\bm1;\tau)}{P_\lambda(\bm1;\tau)}-\frac{P_\mu(x+\bm1;\tau)}{P_\mu(\bm1;\tau)}\in\mathbb F_{\geqslant0}^{\mathbb R},\qquad \forall x\in[0,\infty)^n.$$

2. For some fixed $\tau_0\in[0,\infty]$,

$$\frac{P_\lambda(x+\bm1;\tau_0)}{P_\lambda(\bm1;\tau_0)}-\frac{P_\mu(x+\bm1;\tau_0)}{P_\mu(\bm1;\tau_0)}\geqslant0,\qquad \forall x\in[0,\infty)^n.$$

3. $\lambda$ weakly majorizes $\mu$. The conjecture is presented as the weak-majorization analogue of the preceding Jack conjecture; the supplied text gives no general resolution.