Finiteness conjecture for genera of rational vertex operator algebras

Let $V$ be a rational vertex operator algebra, and let its genus consist of its associated modular braided tensor category together with its central charge. Write

$$\operatorname{gen}(V):=\{\text{isomorphism classes of vertex operator algebras with the same genus as }V\}.$$

Finiteness conjecture. The set $\operatorname{gen}(V)$ is finite.

This asks whether fixing the modular braided tensor category and central charge leaves only finitely many isomorphism classes of rational vertex operator algebras. The source gives no resolution of the conjecture.