Erik Demaine and I also included a proof for $d=2$ in Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Chapter 3. There we asked if there is a planar (non-crossing) linkage that "signs your name" (traces any semi-algebraic region), a question from Don Shimamoto in 2004. This was recently settled positively by Zachary Abel in his Ph.D. thesis:
Abel, Zachary Ryan. "On folding and unfolding with linkages and origami." PhD diss., Massachusetts Institute of Technology, 2016. MIT link.