4 added 45 characters in body

It is wrong but may give someone an idea.

For 1D I would go for +oo+++ooooo+++oo+ (+ is a cell, o is a hole). The key point is that we have more holes than cells but still, each hole requires its own filler. Making the graph P->fillers of holes in P, we get the outbound degree 9 and the inbound degree 8. But for all poliominoes in a cube of size N, their fillers are in the cube of size N+100, so the number of fillers cannot exceed the number of poliominoes noticeably.

In 2D the 7 by 7 square frame has the same properties and is connected..

3 added 76 characters in body

For 1D I would go for +oo+++ooooo+++oo+ (+ is a cell, o is a hole). The key point is that we have more holes than cells but still, each hole requires its own filler. Making the graph P->fillers of holes in P, we get the outbound degree 9 and the inbound degree 8. But for all poliominoes in a cube of size N, their fillers are in the cube of size N+100, so the number of fillers cannot exceed the number of poliominoes noticeably.

In 2D the 7 by 7 square frame has the same properties and is connected..

2 edited body

For 1D I would go for +--+++-----+++--+ oo+++ooooo+++oo+ (+ is a cell, - o is a hole). The key point is that we have more holes than cells but still, each hole requires its own filler. Making the graph P->fillers of holes in P, we get the outbound degree 9 and the inbound degree 8. But for all poliominoes in a cube of size N, their fillers are in the cube of size N+100, so the number of fillers cannot exceed the number of poliominoes noticeably.

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