Generic SNP conjecture for nonsymmetric Macdonald polynomials

Let $\alpha\in\mathbb Z_{\geq0}^n$, and let $E_\alpha(X;q,t)$ be the nonsymmetric Macdonald polynomial. Write

$$E_\alpha(X;q,t)=x^\alpha+\sum_{\beta<_S\alpha}d_{\alpha,\beta}(q,t)x^\beta,$$

where $<_S$ is the ordering generated by the stated transposition and transfer moves. Let $\widehat{\mathcal P}_\alpha$ be the convex hull of all $\beta\in\mathbb Z_{\geq0}^n$ with $\beta\leq_S\alpha$. Generic SNP conjecture. If $\beta\in\widehat{\mathcal P}_\alpha$ and $\beta\in\mathbb Z_{\geq0}^n$, then $\beta\leq_S\alpha$. This would imply that the generic specialization of $E_\alpha$ is SNP; the claim was checked for $n\leq7$ and $|\alpha|\leq7$, but remains open in general.