The rational vertex operator algebra decomposition conjecture

Let $V$ be a rational vertex operator algebra, and let $W_{1},\dots,W_{r}$ be a complete set of representatives of equivalence classes of irreducible $V$-modules. Here $U(V)$ is the associative algebra associated with $V$, and $gl_{J}(W_i)$ denotes the corresponding algebra of $J$-graded endomorphisms of $W_i$. Decomposition conjecture. Then

$$U(V)=gl_{J}(W_{1})\oplus\cdots\oplus gl_{J}(W_{r}).$$

This proposes a complete algebraic decomposition of $U(V)$ indexed by the irreducible modules of a rational vertex operator algebra. The preceding corollary establishes the corresponding quotient decomposition for a finite direct sum of inequivalent irreducible modules, but the displayed equality for $U(V)$ itself is stated here as the paper's concluding conjectural claim.