Jack-parameter monotonicity in the Jack semiring

Let $\lambda$ and $\mu$ be partitions, and let $\mathcal J_C^\sigma(\mathbb R_{\geqslant0})$ be the Jack cone generated by normalized Jack differences with parameter $\sigma$. Fix $0\leqslant\sigma\leqslant\tau\leqslant\infty$. Jack-semiring conjecture. If $\lambda$ majorizes $\mu$, then

$$\frac{P_\lambda(x;\tau)}{P_\lambda(\bm1;\tau)}-\frac{P_\mu(x;\tau)}{P_\mu(\bm1;\tau)}\in\mathcal J_C^\sigma(\mathbb R_{\geqslant0}).$$

This predicts positivity at parameter $\tau$ in the cone generated at every smaller parameter $\sigma$; the supplied text gives no general resolution.