Positivity of Wahl defects and representations from index-one covers

Let \((X,E) \to (V,0)\) be the minimal good resolution of a normal complex surface singularity, with \(E\) reduced. The Wahl defect is \(Δ(V,0) :=h^1(\mathcal{O}_X)-h^1(S_X)+h^1(\bigwedge^2 S_X)\), where \(S_X=Θ_X(-\log E)\). For non-Gorenstein \(\mathbb{Q}\)-Gorenstein singularities \((V,0)\), we p