The Kazhdan-Lusztig category of W-algebras of simply-laced Lie algebras at irrational levels

Let $\mathfrak{g}$ be a simple, simply-laced Lie algebra and $f\in \mathfrak{g}$ nilpotent. The Kazhdan-Lusztig category of the W-algebra $W^{\kappa }(\mathfrak{g},f)$ associated with $(\mathfrak{g},f)$ at level $\kappa \in \mathbb{C}$ is obtained from the Kazhdan-Lusztig category of the affine vertex algebra $V^{\kappa }(\mathfrak{g})$ via the quantum Hamiltonian reduction associated with $f$. We show that this is a braided tensor equivalence for any $f$ and any irrational level $\kappa \in \mathbb{C}\mathrm{\backslash }\mathbb{Q}$.