Positivity and unimodality for inflated characters of Weyl groups

Let $G$ be a connected complex reductive group with Weyl group $W$, let $\psi\in\operatorname{Irr}(W)$ be inflated from a quotient

$$\varphi\colon W\to\Omega,$$

and let $\alpha_{\psi,G}^{z}$ be the Laurent polynomial associated with $\psi$ and $z\in W$. Say that $(\vartriangle)_{\psi,G}^{z}$, $(\pm)_{\psi,G}^{z}$, and $(+)_{\psi,G}^{z}$ denote the three positivity or unimodality properties defined in the paper. The new conjecture. If $\Omega$ is isomorphic to a product of symmetric groups, then $(\vartriangle)_{\psi,G}^{z}$ and $(\pm)_{\psi,G}^{z}$ hold for every $z\in W$. Moreover, if $\varphi$ does not descend to the $G_2$-to-$A_2$ quotient, then $(+)_{\psi,G}^{z}$ holds for every $z\in W$.

The conjecture proposes that compatibility between inflation maps and the Springer correspondence lifts Haiman's positivity and unimodality from symmetric groups to other Weyl groups. The supplied text gives no resolution and records it as a new conjecture.