The conjecture on the image of the conformal embedding map

Let $n$ be as in the conformal embedding under consideration, and let $V^{-n}(sl(n))$ be the universal affine vertex algebra at the critical level. Let $S^{(2)}=S,S^{(3)},\dots,S^{(n)}$ be the Feigin–Frenkel central elements, and define

$$s_k=S^{(k)}\otimes {\bf 1}-{\bf 1}\otimes S^{(k)}.$$

Set

$$\mathcal V(sl(n)\times sl(n))=\left(V^{-n}(sl(n))\otimes V^{-n}(sl(n))\right)\big/\langle s_2,s_3,\cdots,s_n\rangle.$$

The conjecture on the image of the conformal embedding map. The image of the homomorphism $\Phi$ is isomorphic to

$$\operatorname{Im}(\Phi)\cong\mathcal V(sl(n)\times sl(n)).$$

This conjecture proposes that the only relations defining the image of the conformal embedding map are the equalities of the corresponding Feigin–Frenkel central elements in the two tensor factors. The preceding proposition shows that the image is a quotient of a related algebra obtained by imposing the relation for the quadratic central element, but the full asserted isomorphism is left as a conjecture here.