Grothendieck–Verlinde fusion ring isomorphism conjecture for bosonic ghosts

Let $\mathbin{\times}$ denote the fusion product on the $\mathbb{Z}$-span of the indecomposable $\mathfrak{G}$-modules, where addition is direct sum. Assume that fusing with any given $\mathfrak{G}$-module defines an exact functor, so that the fusion product descends to a well-defined product $\mathbin{\boxtimes}$ on the Grothendieck group:

$$\bigl[ \mathcal{M} \bigr] \mathbin{\boxtimes} \bigl[ \mathcal{N} \bigr] = \bigl[ \mathcal{M} \mathbin{\times} \mathcal{N} \bigr].$$

Grothendieck–Verlinde fusion ring isomorphism conjecture. The product on the resulting Grothendieck ring may be identified with the Verlinde product under the group isomorphism $\bigl[ \mathcal{M} \bigr] \mapsto \mathrm{ch} \bigl[ \mathcal{M} \bigr]$; equivalently, this map constitutes an isomorphism between the Verlinde and Grothendieck fusion rings.

The conjecture would identify the fusion rules obtained from the representation-theoretic Grothendieck ring with the Verlinde product computed from characters. The exactness assumption and the claimed compatibility remain unproved in the source.