The paranormality of products and their subsets

A topological space is called paranormal if any countable discrete system of closed sets $\{D_n{:}n=1,2,3,\ldots\}$ can be expanded to a locally finite system of open sets $\{U_n{:}n=1,2,3,\ldots\}$, i.e., $D_n$ is contained in $U_n$ for all $n$ and $D_m\cap U_n\neq\emptyset$ if and only if $D_m=D_n$. It is proved that if $X$ is a countably compact space whose cube is hereditarily paranormal, then $X$ is a metrizable space.