### A lower bound of the upper bound from polyomino-covering in [S]

In [S], Soifer derives .
As he mentioned, one can improve the covering construction. Holding the square of side length in the lower left corner, putting a square of side length in the upper right corner, covering the remaining uncovered area by 2 polyomino-coverings of rectangles of sides by , removing useless unit squares in polyominos, we get a lower bound for the rhs of that inequality:

Denote by the minimal value of this expression when varying from 2 to .
Results of computer calculations:
iff or .
For growing (checked up to ), for the lowest optimal , seems to converge to 1, and seems to converge to 3/4.