Maximum number of colors for an optimal tiling which guarantees infinite paths

Question

This question is a more applicable version of the question I've asked in mathexchange recently:

What is the maximum number of colors we can use to color $N^2$ square sub-tiles of $N×N$ square block tile, for which there exist an optimal tiling of whole plane with these identical tile blocks(only four states of rotating $90$ degree blocks can be used)that have at least a two-head infinite paths for each color?

Note: A two-head infinite path of one coloured sub-tiles consist of elements of sub-tiles with the same color which for each element in the path there exist two another non-adjacent elements of the path from its 8 adjacent sub-tiles squares.(also it does not intersect with itself for example does not make circle)

Answer 1

If you do not allow the square block tiles to be rotated, the answer is $N$.

1. The number is at least $N$ because with $N$ colours you can easily can construct a square block tile such that there is a path for each colour (e.g. use the same colour for all squares in one row).
2. The number is at most $N$ because each colour needs to cover at least $N$ squares in the square block tile. You can proof this using the following argument: Assume there exists a such tiling, then any $N\times N$ square in this tiling gives a valid square block tile. Moreover the number of sub-tiles having a fixed colour does not depend on the $N\times N$ square you choose. Take a segment of $N$ squares in infinite path of a fixed colour. This segment fits into an $N\times N$ square. Thus the colour must be used in at least $N$ tiles in any square block tile.

I did not think about the case, where you can rotate the block tiles, yet.