Truncated Weyl module–fusion product isomorphism conjecture

Let $P^+$ be the set of dominant integral weights. For $\lambda\in P^+$ and $N\geq1$, let $\operatorname{W}(\lambda,N)$ be the truncated Weyl module for $\mathbf U(\mathfrak g\otimes\mathbb C[t]/t^N)$, and let $P^+(\lambda,N)$ be the set of $N$-tuples of dominant integral weights summing to $\lambda$. Suppose $|\lambda|\geq N$, and let $\blambda=(\lambda_1,\dots,\lambda_N)$ be the unique maximal element of $P^+(\lambda,N)$. Truncated Weyl module–fusion product conjecture. There is an isomorphism of $\mathbf U(\mathfrak g\otimes\mathbb C[t]/t^N)$-modules

$$\operatorname{W}(\lambda,N)\cong V(\lambda_1)*\cdots*V(\lambda_N).$$

The conjecture connects truncated Weyl modules with fusion products of irreducible finite-dimensional $\mathfrak g$-modules. The paper presents evidence for it and computes a PBW-type basis in the case $\mathfrak g=\mathfrak{sl}_2$; no general resolution is supplied.