I came up with the following conjecture the other day, and was wondering if the result was well-known or even true:
Define $f(P)$ for a polyomino $P$ (without holes) to be the least number of total pieces in which two copies of $P$ can be split such that these pieces can be rearranged to form a copy of $P$ scaled up by $\sqrt{2}$.
Conjecture: For a polyomino $P$ with area $A$, we have $f(P)\le A+3$.
This bound seems very sharp for small polyominoes. So far I've shown that $f(P)\le A+3$ holds for all rectangle polyominoes with a relatively simple construction. Is this true in general? Or is there any weaker non-trivial (i.e., better than $f(P)\le 4A$) bound we can prove in the general case or some specific case?