Is there an arrangement of finitely many axes-parallel squares in the plane, of $k$ different colors, such that:

* The squares of each color are pairwise-disjoint;
* Each square overlaps at least $4$ squares of some other color (e.g. each red square overlaps $4$ green or $4$ blue squares; each blue square overlaps $4$ red or $4$ green squares; etc.)?

For $k=2$ colors, [Barry Cipra proved that the answer is no](https://math.stackexchange.com/a/646208/29780). Can his argument (or another one) be extended to $k>2$?

EDIT:

The beautiful constructions by Manfred Weis show that, for $k\geq 4$, there exists an arrangement in which each square overlaps at least 4 other squares that may be of different colors (e.g. a red square may overlap two green and two blue squares). But my intention (which was probably not clear enough) was that each square should overlap at least 4 other squares of the same color (e.g. a red square should overlap 4 green squares, or 4 blue squares).