Rationality and Zhu algebra semisimplicity conjecture for vertex operator algebras

Let $V$ be a simple vertex operator algebra, and let $A(V)$ denote its Zhu algebra.

Rationality–semisimplicity conjecture. $V$ is rational if and only if $A(V)$ is semisimple.

This conjecture extends the preceding result that rationality and semisimplicity of $A(V)$ are equivalent under suitable finiteness assumptions, such as $C_1$-cofiniteness or $C_2$-cofiniteness. It asks whether those assumptions can be removed for every simple vertex operator algebra.