On the Chatyrko-Hattori solution of de Groot's question

Let Zn=([0,1]n×(0,1])∪(∂([0,1]n)×{0}). De Groot asked: Is cmpZn⩾n for every n? It is known that the answer is yes for n=1 and 2. V.A. Chatyrko and Y. Hattori [Fund. Math. 172 (2002) 107-115] showed that the answer is no for n⩾5. It is shown that the answer is also no for n=4. The question is unresolved for n=3.