Completeness conjecture for Conway and umbral K3 twining genera

Let $\mathcal{T}(\Pi)$ be a $K3$ nonlinear sigma model with symmetry group $G$, let $\Gamma_G$ be its co-invariant lattice, and let $\Phi(N)$ denote the corresponding set of Conway or umbral twining functions for a Niemeier lattice $N$. Completeness conjecture for K3 twining genera. There exists at least one Niemeier lattice $N$ such that $\Gamma_G$ can be embedded in $N$, $G\subseteq G_N$, and, for every $g\in G$, the twining genus ${\cal Z}_g$ coincides with an element of $\Phi(N)$. Equivalently, for each $K3$ nonlinear sigma model, all physical twining genera should be contained in $\Phi(N)$ for some Niemeier lattice. The conjecture is supported by known geometric and Landau--Ginzburg examples and by consistency with parity, but remains open.