Consider arrangements of triominoes and tetrominoes on a grid where each polyomino touches polyominoes of total area 4 times their own. (Here "on a grid" means the vertices of the underlying squares have integer coordinates, and "touch" means share positive perimeter.)
It is not too hard to find a disconnected collection a triominoes and tetrominoes with this property, where each type of polyomino touches 4 others of the same area:
Is there a connected example? If so, it is not hard to see that each triomino would need to touch exactly 3 tetrominoes, and each tetromino would need to touch exactly 1 tetromino and 4 triominoes.