Katetov's cube theorem and seminormal functors

Assuming the Jensen’s principle we costructed an example of nonmetrizable compact X with the following properties: for every seminormal functor F and n ≥ 2 the space F n(X) \ F n-1(X) is perfectly normal and F n(X) is hereditarily separable. In particular, for every n > 2 X n is hereditarily separable and X n \ Δ n is perfectly normal, where Δ n is the generalized diagonal of X n.