Friedan's reconstruction conjecture for self-dual vertex operator algebras

Let $U$ and $V$ be self-dual VOAs of CFT type, and let $\mathcal{Z}_{U,g}$ and $\mathcal{Z}_{V,g}$ denote their genus-$g$ partition functions.

Reconstruction conjecture. If

$$\mathcal{Z}_{U,g}=\mathcal{Z}_{V,g}$$

for every $g$, then $U$ and $V$ are isomorphic.

This is the VOA analogue of Friedan's proposal that a field theory is determined by its restriction to closed surfaces. The source gives no resolution of the conjecture.