Pairwise Touching Polyominoes In Boxes

We say a collection of polyominoes is pairwise touching if every polyomino touches every other along at least one face. If we cut up an n x m x p box into polyominoes, what is the maximum number of pairwise touching polyominoes we can get? The tables below show the best-known bounds. Click on the links for pictures.

Pairwise Touching Polyominoes in n x m x 1 Box
n \ m 1 2 3 4 5
1 1 2 2 2 2
2 2 3 3 3 3
3 2 3 4 4 4
4 2 3 4 4 4
5 2 3 4 4 4

Pairwise Touching Polyominoes in n x m x 2 Box
n \ m 1 2 3 4 5
1 2 3 3 3 3
2 3 4 4 4 4
3 3 4 5 6 6
4 3 4 6 7 7,8
5 3 4 6 7,8 7,10

Pairwise Touching Polyominoes in n x m x 3 Box
n \ m 1 2 3 4 5
1 2 3 4 4 4
2 3 4 5 6 6
3 4 5 5 6 7
4 4 6 6 8 8,10
5 4 6 7 8,10 9,13

Pairwise Touching Polyominoes in n x m x 4 Box
n \ m 1 2 3 4 5
1 2 3 4 4 4
2 3 4 6 7 7,8
3 4 6 6 8 8,12
4 4 7 8 9,12 9,16
5 4 7,8 8,12 9,16 10,18