Seminormal functors and paranormality

It is known that if a space $\mathcal{F}(X)$ is hereditarily paranormal for a paracompact $p$-space $X$ and normal functor $\mathcal{F}$ of degree $\ge 3$ in the category $\mathcal{P}$ of paracompact $p$-spaces and their perfect maps, then $X$ is metrizable. In this paper, a generalization of this theorem is proved for seminormal functors in the category $\mathcal{P}$.