  In this paper, we prove that in forcing extensions by a poset with finally
property K over a model of GCH+$\square$, every compact sequentially compact
space is weakly pseudoradial. We also prove the following assuming
$\mathfrak{s}\leq \aleph_2$: (i) if $X$ is compact weakly pseudoradial, then
$X$ is pseudoradial if and only if $X$ cannot be mapped onto
$[0,1]^\mathfrak{s}$; (ii) if $X$ and $Y$ are compact pseudoradial spaces such
that $X\times Y$ is weakly pseudoradial, then $X\times Y$ is pseudoradial.
These results add to the wide variety of partial answers to the question by
Gerlits and Nagy of whether the product of two compact pseudoradial spaces is
pseudoradial.
