FIRST THOUGHT:

King can hit at most 8 pieces = 8
Queen can hit at most 8 pieces = 8
Each rook can hit at most 4 pieces = 2x4 = 8
Each bishop can hit at most 4 pieces =2x4 = 8
Each knight can hit at most 8 pieces = 2x8 = 16
Each pawn can hit at most 2 pieces = 8x2 = 16

So obviously the true answer can't possibly be above 64 connections.

Edit to add SECOND THOUGHT:
We can reduce this further. From all columns in which there are pieces, one piece atleast occupies the leftmost column, atleast one piece the rightmost column. These pieces will have their potential attacks limited, atleast by one (if these pieces are pawns), so the true answer can't possibly be above 62 connections.

Edited to fix bad arithmetic.