Normal functors on the category of paracompact $p$-spaces and their perfect mappings

It is proved that for any paracompact $p$-space $X$ and any normal functor ${\mathscr F}$ of degree $\geq3$ in the category $\mathscr{P}$ the hereditary normality of the space $\mathscr{F}(X) \setminus X$ implies the metrizability of $X$.