  This is the second paper in the series devoted to the study of the dimer
model on t-embeddings of planar bipartite graphs. We introduce the notion of
perfect t-embeddings and assume that the graphs of the associated origami maps
converge to a Lorentz-minimal surface $\mathrm{S}_\xi$ as $\delta\to 0$. In
this setup we prove (under very mild technical assumptions) that the gradients
of the height correlation functions converge to those of the Gaussian Free
Field defined in the intrinsic metric of the surface $\mathrm{S}_\xi$. We also
formulate several open questions motivated by our work.
