[Math] Is the ratio Perimeter/Area for a finite union of unit squares at most 4

Question

Update: As I have just learned, this is called Keleti's perimeter area conjecture.

Prove that if H is the union of a finite number of unit squares in the plane, then the ratio of the perimeter and the area of their union is at most four.

Remarks. If the squares must be axis-aligned, then this is easy to prove. If we replace unit squares with unit circles, then the statement is sharp and true for two (instead of four). The best known bound (to me) is 5.551… by Zoltán Gyenes. There is much more about the problem in his master thesis which you can find here.

Answer 1

Disproved by Viktor Kiss and Zoltán Vidnyánszky, see http://arxiv.org/abs/1402.5452