Suppose we were to ignore, at least for now, the curviness of the segments and imagine them as straight diagonals across a unit square. Then the largest region we could contain would have area 36 (top), comprising 18 squares of area 2.

To draw a “curvy” grid containing the same area, we would simply need to replace the 22 perimeter segments of such a grid with 11 “outward”-bulging quarter circles segments and 11 “inward”-bulging segments.

In a similar vein, in order to get an area of 32 we need to base our curve off a region comprising 16 of the 18 squares. Depending on the choice of squares to omit, the perimeter of the remaining region is either 18, 20, 22, or 24. (The example above shows one way to attain a 24-segment perimeter.)

Counting all the cases for removing 2 boxes, and also accounting for mirror symmetry, the total number of grids of area 32 is 2*Binomial[18,9] + 36*Binomial[20,10] + 56*Binomial[22,11] + 16*Binomial[24,12] = 89,519,144.

Congrats to the solvers that successfully computed the number of arrangements!