Regularity of the fan from the discriminantal hyperplane arrangement

Prove that the fan Σ defined by the linearized discriminantal hyperplane arrangement 𝓗^{lin}_Φ = {h_α | α ∈ Φ} is regular, thereby implying that the associated toric variety X_Σ is smooth.

Background

The paper constructs a toric variety X_Σ from the linearized discriminantal arrangement associated with hypertoric varieties and seeks to compactify the parameter space for quantum multiplication.

Smoothness of X_Σ (equivalently, regularity of Σ) is assumed for subsequent arguments, but this regularity is not established in general; proving it is posed as a conjecture and would validate the smoothness hypothesis broadly.