The generic rigidity characterization by pseudo-triangulation straightening

Let $G$ be a plane graph. A plane graph is generically rigid if its generic realizations are infinitesimally rigid, and it can be straightened as a pseudo-triangulation if its embedding admits a straight-line realization whose faces are pseudo-triangles. The generic rigidity characterization. The following conditions are equivalent:

1. $G$ is generically rigid.
2. $G$ can be straightened as a pseudo-triangulation.

This conjecture seeks a complete characterization of the plane graphs that admit pseudo-triangulation realizations. The paper establishes the result for several important subclasses, including plane Laman graphs, rigidity circuits, and certain Laman-plus-one and Laman-plus-two graphs, while leaving the general case open.