Ollinger's conjecture on undecidability of 8-, 9-, and 10-polyomino tiling

For a fixed positive integer $k$, the $k$-polyomino tiling problem asks whether there is an algorithm deciding whether a set of $k$ polyominoes can tile the plane by translated copies from the set. Ollinger's conjecture. The $k$-polyomino tiling problem is undecidable for $k=8,9,10$. Ollinger had already proved undecidability for $k=11$, while the conjecture was motivated by the existence of an aperiodic set of $8$ polyominoes. The present paper claims to prove the conjecture by establishing the $k=8$ case; the cases $k=9,10$ then follow by adding redundant polyominoes.