A computer search led to the following 16x16 example with a 7/2 rectangle in the middle (scale everything by 1/12):
0 -3 3 0 0 0 0 0 0 0 0 -6 6 0 0 0
-1 1 -3 0 3 3 -3 0 0 -3 0 6 0 -3 0 0
4 2 0 0 -12 9 -3 0 0 0 9 -12 0 -3 6 0
-6 0 0 0 6 -12 12 -6 0 12 -12 3 3 2 -2 -6
6 0 0 0 6 0 -12 6 12 -12 0 3 2 -1 -4 12
-3 0 0 0 -3 2 -2 0 -6 6 -6 12 -5 -7 0 0
0 0 0 0 3 4 -4 -6 0 -12 12 -6 0 12 0 -6
0 0 0 0 -6 -12 12 6 12 12 -12 -6 0 0 0 0
0 9 -6 0 6 2 -2 -4 -2 -10 10 0 5 -9 -2 0
3 -3 0 0 0 4 -4 -2 -4 4 -4 6 0 4 2 -6
0 -6 6 0 0 0 -12 6 12 -12 0 -3 -2 1 4 6
-6 0 0 0 6 -12 12 2 -8 12 -12 3 3 0 0 -6
8 0 -6 4 -12 9 3 -5 -1 0 9 -6 -4 4 0 3
0 0 0 -4 4 0 0 3 -7 0 3 1 -2 -1 -4 3
-2 0 3 -1 0 3 0 -1 2 0 -6 -1 7 0 0 0
-3 0 6 1 0 -4 0 1 -1 -3 3 0 0 0 0 0
Update : using Leonid's remark on imposing symmetry w.l.o.g., here is a smaller and symmetric 13x16 example with a 7/2 rectangle (scale by 1/4):
0 0 0 0 1 -1 0 0 0 0 -1 1 0 0 0 0
1 0 0 -2 -2 4 -1 0 0 -1 4 -2 -2 0 0 1
-2 0 0 2 0 -4 4 -1 -1 4 -4 0 2 0 0 -2
1 0 0 0 0 0 -4 3 3 -4 0 0 0 0 0 1
0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0
0 0 0 0 2 2 -2 -1 -1 -2 2 2 0 0 0 0
0 0 0 0 -2 -4 4 3 3 4 -4 -2 0 0 0 0
0 0 0 0 2 2 -2 -1 -1 -2 2 2 0 0 0 0
0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0
1 0 0 0 0 0 -4 3 3 -4 0 0 0 0 0 1
-2 0 0 2 0 -4 4 -1 -1 4 -4 0 2 0 0 -2
1 0 0 -2 -2 4 -1 0 0 -1 4 -2 -2 0 0 1
0 0 0 0 1 -1 0 0 0 0 -1 1 0 0 0 0
I also tried to find an example with a ratio better than 7/2, keeping the symmetry and a 1x4 rectangle in the middle, but could not find any (the largest size I am able to check is 33x38).