The fusion-product level-scaling conjecture for simple Lie algebras

Let ${\mathfrak g}$ be a simple Lie algebra, let $\lambda^1,\dots,\lambda^n$ be weights, let $V(\lambda^i)$ be the corresponding finite-dimensional ${\mathfrak g}$-modules, and let $V_1*\dots*V_n(z_1,\dots,z_n)$ denote their fusion module at pairwise distinct parameters. For a graded module $M$, write $M^{[k]}$ for the level-scaling construction used in the paper. Fusion-product level-scaling conjecture. For any simple ${\mathfrak g}$ and any weights $\lambda^1,\dots,\lambda^n$,

$$\left(V(\lambda^1)*\dots*V(\lambda^n)(z_1,\dots,z_n)\right)^{[k]}\cong V(k\lambda^1)*\dots*V(k\lambda^n)(z_1,\dots,z_n).$$

The preceding proposition gives a quotient from the right-hand side onto the left-hand side; the conjecture asserts that this quotient is an isomorphism.