Erik Demaine and I also included a proof for $d=2$ in [Geometric Folding Algorithms:
Linkages, Origami, Polyhedra](http://www.gfalop.org/), 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](https://dspace.mit.edu/handle/1721.1/107547).