FLM uniqueness conjecture for the moonshine module vertex operator algebra

Let $V$ be a vertex operator algebra of central charge $24$ with no nonzero elements of weight $1$. Assume that every irreducible $V$-module is equivalent to $V$ as a $V$-module and that every $V$-module is completely reducible.

Moonshine uniqueness conjecture. Then $V$ is isomorphic to the moonshine module vertex operator algebra.

This conjecture would characterize the moonshine module from its central charge, vanishing weight-one space, and complete reducibility properties; the source notes that it would also give a definition of the Monster group as its automorphism group. No resolution is supplied.