Twined elliptic genus conjecture for K3 sigma models determined by positive planes

Let $\Pi=P_X\oplus P_Z$ be the positive plane determining a supersymmetric non-linear K3 sigma model, and let $G_\Pi$ be the corresponding group of automorphisms. For $g\in G_\Pi$, let $\phi_g$ be the Jacobi form attached to the corresponding derived autoequivalence. Twined elliptic genus conjecture. The twined elliptic genus attached to the supersymmetry-preserving automorphism $g\in G_\Pi$ of the supersymmetric non-linear K3 sigma model determined by $\Pi=P_X\oplus P_Z$ coincides with $\phi_g$. This is supported by the explicit comparison of the paper's functions with K3 sigma-model twining genera, but the identification for all such models and automorphisms remains conjectural.