On the Chatyrko–Hattori solution of de Grootʹs question

Let Z n = ( [ 0 , 1 ] n × ( 0 , 1 ] ) ∪ ( ∂ ( [ 0 , 1 ] n ) × { 0 } ) . De Groot asked: Is c m p Z n ⩾ 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 .