Realization conjecture for all Conway and umbral twining functions

Let $N$ range over the 24 Niemeier lattices and let $\Phi(N)$ be the associated set of Conway or umbral twining functions. Realization conjecture. For every element $\phi\in\Phi(N)$, there exists a $K3$ nonlinear sigma model $\mathcal{T}$ with a symmetry $g$ such that

$$\phi={\cal Z}_g(\mathcal{T}.)$$

Thus every function produced by Conway or umbral moonshine should arise as a physical twining genus of a $K3$ nonlinear sigma model. The source presents this as a converse to the completeness conjecture; it remains unproved.