Jack differences in the Muirhead semiring

Fix $\tau_0\in[0,\infty]$, and let $\lambda$ and $\mu$ be partitions. Let $\mathcal M_S(\mathbb R_{\geqslant0})$ denote the Muirhead semiring over the nonnegative reals. Muirhead-semiring conjecture. If $\lambda$ majorizes $\mu$, then

$$\frac{P_\lambda(x;\tau_0)}{P_\lambda(\bm1;\tau_0)}-\frac{P_\mu(x;\tau_0)}{P_\mu(\bm1;\tau_0)}\in\mathcal M_S(\mathbb R_{\geqslant0});$$

and conversely. This strengthens the Jack positivity conjecture by replacing positivity with membership in the Muirhead semiring. The supplied status evidence says that the analogous Muirhead-cone assertion is false, but does not refute this semiring statement.