Can every abstract simplicial complex whose geometric realization is homeomorphic to $\mathbb{R}^2$ be realized by a rectilinear triangulation of the Euclidean plane? Alternatively put, can a curvy (honest) triangulation of $\mathbb{R}^2$ be straightened?

In a related question (Euclidean triangulation of the plane with degree 7 at each vertex.) the answers show in effect that a geodesic triangulation of the hyperbolic plane is combinatorially equivalent to a rectilinear triangulation of the Euclidean plane. It would therefore suffice to realize an abstract triangulation in the hyperbolic plane.