Conjecture on elliptic root systems for Weyl group Jacobi orbit spaces

Let $W$ be a Weyl group, let $\Omega$ be the relevant domain, and let $J(W)$ be the Jacobi group acting on it. Write $\mathcal{R}_W$ for the root system of $W$, let $\mathcal{R}_W^{irreg}$ denote the irregular part of the elliptic $\vee$-system, let $\mathfrak{U}$ be the corresponding configuration, and let $\Delta$ be a weight vector. The dual prepotential is expressed in the form

, with $h^\vee_\mathfrak{U}=0$. **Elliptic $\vee$-system conjecture.** For the Jacobi group orbit space $\Omega/J(W)$, the dual prepotential takes the form

with $h^\vee_\mathfrak{U}=0$. Furthermore, $\mathfrak{U}=\mathcal{R}_W\cup\mathcal{R}_W^{irreg}$, or its dual, where

$$\mathcal{R}^{irreg}_W=\{w(\Delta)\mid w\in W\}\quad\text{or}\quad\mathcal{R}^{irreg}_W=\{\pm w(\Delta)\mid w\in W\}.$$

Here the conjecture proposes a uniform description of the dual prepotentials and the irregular components for Jacobi group orbit spaces associated with Weyl groups; its general status is not specified in the source.