Gaussian free-field limits of weighted integer Lipschitz heights

We prove a Gaussian free field scaling limit for weighted integer Lipschitz heights on the triangular lattice. With zero boundary values and a factor x for each edge on which the height changes, the field converges after division by a positive constant depending only on x to the zero-Dirichlet Gaussian free field, for every fixed $x\in [1/\sqrt{2},1]$. The convergence holds as a random distribution on every bounded C${}^2$ Jordan domain under inside lattice approximations with uniformly convergent boundary parametrizations. This includes the uniform height model and the predicted critical endpoint.