The fusion–spectral-flow compatibility conjecture for affine $\widehat{\mathfrak{sl}}(2)$-modules

Let $V$ and $W$ be admissible level $k$ modules of $\widehat{\mathfrak{sl}}(2)$, let $s,t\in\mathbb Z$, and let $k\in\{-1/2,-4/3\}$. Write $V^s$ for the module obtained from $V$ by spectral flow by $s$, and similarly for $W^t$. Fusion–spectral-flow compatibility conjecture.

$$V^s\times W^t=(V\times W)^{s+t}.$$

This compatibility is an important expected property in the representation theory of affine vertex algebras and is used in the construction of the extended algebras and their modules. The source notes consistency with the Verlinde formula for admissible level $\widehat{\mathfrak{sl}}(2)$, but the conjecture is not proved there.