Say that we have a polyomino $P$ which tiles the plane. In may cases, it can do so by forming a two-tile "patch" which tiles the plane. For instance, with the T pentomino:
Is there always a way to take the disjoint union of two congruent copies of $P$ to form a polyomino $Q$ which also tiles the plane?
This feels intuitively false to me - it doesn't feel like there should be any reason for it to hold, even in the case of periodic tilings (once the period is large enough, anyway). However, it holds for every polyomino on up to $9$ cells, so I've struggled to locate a counterexample.
Can anyone provide a counterexample to this claim (or, if my intuition is completely off, a proof)?
Note that one obstacle is the Conway criterion; any tile which satisfies it will generate such a $Q$.
This should be a much easier-to-disprove question than this one, which would require the discovery of a connected aperiodic prototile at a minimum.