The Katětov property for finite degree seminormal functors

A seminormal functor $\mathscr F$ enjoys the Katětov property ($K$-property) if for every compact set $X$ the hereditary normality of $\mathscr F(X)$ implies the metrizability of $X$. We prove that every seminormal functor of finite degree $n>3$ enjoys the $K$-property. On assuming the continuum hypothesis ($CH$) we characterize the weight preserving seminormal functors with the $K$-property. We also prove that the nonmetrizable compact set constructed in [1] on assuming $CH$ is a universal counterexample for the $K$-property in the class of weight preserving seminormal functors.