Existence and finite-dimensionality conjecture for Griess algebras

Let $A$ be a commutative algebra with a symmetric invariant bilinear form. An OZ vertex algebra is a graded vertex algebra $V=\bigoplus_{i\in\mathds Z}V_i$ such that $\dim V_0=1$ and $V_i=0$ for $i=1$ or $i<0$.

Griess algebra realization conjecture. There is an OZ vertex algebra $V$ such that

$$A=V_2.$$

Moreover, if $\dim A<\infty$, then $V$ can be chosen so that

$$\dim V_n<\infty\quad\text{for }n=3,4,5,\ldots.$$

The preceding theorem proves the corresponding realization with $A$ embedded in $V_2$ for a non-degenerate form, and also gives simplicity, but does not establish equality $A=V_2$ or the asserted finite-dimensionality of the higher graded pieces. The authors suggest that the conjecture should hold; no resolution is supplied here.