The terminating Weyl-antisymmetry conjecture for the first eigenfunction

Let $W(A_{n-1})$ be the Weyl group of type $A_{n-1}$ acting through the representation $\pi_m$ on the polynomial space ${\cal P}_n$, and let $f_{0,\ldots,0}(\zeta_1,\ldots,\zeta_n)$ be the first eigenfunction of $I(\alpha)$ or $D$. If $m$ is a positive integer and $t=q^m$, then

$$\prod_{k=1}^{n}\zeta_k^{(n-k)m}f_{0,\ldots,0}(\zeta_1,\ldots,\zeta_n)\in{\cal P}_n.$$

Weyl-antisymmetry conjecture. This polynomial satisfies

$$\pi_m(\sigma)\cdot\prod_{k=1}^{n}\zeta_k^{(n-k)m}f_{0,\ldots,0}(\zeta_1,\ldots,\zeta_n)=-\prod_{k=1}^{n}\zeta_k^{(n-k)m}f_{0,\ldots,0}(\zeta_1,\ldots,\zeta_n)$$

for every $\sigma\in W(A_{n-1})$.

The claim is proved for $n=3$ conditional on the corresponding eigenfunction formula and supported by partial calculations for $n=4$; the general case remains open.