"Heiko Harborth raised the question of whether every planar graph has a straight line representation in which all edge lengths are integers. The answer remains unknown as of 2009."
The reference is this:
Kemnitz, Arnfried, and Heiko Harborth. "Plane integral drawings of planar graphs." Discrete Mathematics 236.1 (2001): 191-195. (Elsevier link)
It is known that every $n$-vertex planar graph has a planar straight-line grid drawing on an $O(n) \times O(n)$ grid. But of course integer vertex coordinates do not imply integer edge lengths!
The Kemnitz-Harborth article concludes with this sentence:
"It seems to be a very hard problem to prove that every pentagon with sides of integral length contains an inner point with rational distances to its vertices."
Here are my questions:
Q. Have there been advances on this problem in the last five years? Perhaps for special cases beyond cubic graphs? Any progress on the specific pentagon problem identified above?
Addendum. User fidbc cited two recent papers exhibiting progress:
Biedl, Therese C. "Drawing some planar graphs with integer edge-lengths." CCCG. 2011.
She shows that planar bipartite, series-parallel graphs, graphs with aboricity $2$, outerplanar graphs, and—most interestingly—that every planar graph of degree at most four, but which is not $4$-regular, has an integral embedding.
Sun, Timothy. "Drawing some $4$-regular planar graphs with integer edge lengths." CCCG. 2013.
Timothy extends Therese's work and captures various families of (but not all) $4$-regular planar graphs.
Finally, TMA's mentions the four-square-corners problem, which is described in this MathWorld article, which says, "J.H. Conway and R. Guy found an infinite numbers of solutions to the problem of three such distances being integers."