The fusion conjecture for higher-level Weyl modules

Let $\lambda^1,\dots,\lambda^n$ be weights, let $W^1(\lambda^i)$ be the level-one Weyl modules, and let $W^1(\lambda^i)^{[k]}$ denote their level-$k$ constructions. Let $*$ denote the fusion product at parameters $z_1,\dots,z_n$. Higher-level Weyl fusion conjecture. One has

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

In particular, the left-hand side is independent of $z_1,\dots,z_n$. The source motivates this statement using the quotient result for fusion products and the associated graded construction.