CGS and Muirhead conjecture for Macdonald polynomials

Let $\lambda$ and $\mu$ be partitions with $|\lambda|=|\mu|$. Let $P_\lambda(x;q,t)$ be the Macdonald polynomial, $\mathcal M_S(\mathbb F_{\geqslant0})$ the Muirhead semiring over the Macdonald positivity cone, and let $\mathbb F_{\geqslant0}^{\mathbb R}$ be its real extension. CGS–Muirhead conjecture for Macdonald polynomials. The following are equivalent:

$$\frac{P_\lambda(x;q,t)}{P_\lambda(\bm1;q,t)}-\frac{P_\mu(x;q,t)}{P_\mu(\bm1;q,t)}\in\mathcal M_S(\mathbb F_{\geqslant0}).$$

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

3. For some fixed $q_0,t_0\in(1,\infty)$, the same difference at $(q_0,t_0)$ is nonnegative for every $x\in[0,\infty)^n$.
4. For some fixed $q_0,t_0\in(1,\infty)$, the same difference at $(q_0,t_0)$ is nonnegative for every $x\in(0,1)^n\cup(1,\infty)^n$.
5. $\lambda$ majorizes $\mu$. The supplied text proves implications between consecutive assertions and special cases, but not the general equivalence.