Associated varieties of simple affine vertex algebras at rational levels

We present a conjecture for associated varieties of simple affine vertex algebras $L_k(\mathfrak{g})$ attached to a simple Lie algebra $\mathfrak{g}$ of simply-laced type and any rational level $k$ greater than the critical level. The key new ingredient compared to the integral case is the covering duality map introduced by Gao-Liu-Lo-Shahidi. We provide evidence for the conjecture.