Universal twined elliptic genus conjecture for K3 sigma models

Let $\Lambda$ be the Leech lattice, and let $V^{s\natural}$ be the canonically-twisted module considered in the paper. A supersymmetry-preserving automorphism of a supersymmetric non-linear K3 sigma model is an automorphism whose action preserves the supersymmetry. Universal twined elliptic genus conjecture. The twined elliptic genus attached to any supersymmetry-preserving automorphism of a supersymmetric non-linear K3 sigma model coincides with $\phi_g$ for some $g\in \operatorname{Co}_0$ fixing a $4$-dimensional subspace of $\Lambda\otimes_{\mathbb Z}\mathbb R$. The conjecture proposes that the $\operatorname{Co}_0$-module $V^{s\natural}$ universally recovers twined K3 elliptic genera; the paper reports agreement with all cases available from explicit computations, but the general assertion remains open.