Box and nabla products that are D-spaces

A space $X$ is $D$ if for every assignment, $U$, of an open neighborhood to each point $x$ in $X$ there is a closed discrete $D$ such that $\bigcup \{U\left(x\right):x\in D\}=X$. The box product, $\square X^{\omega }$, is $X^{\omega }$ with topology generated by all ${\prod }_n{U}_n$, where every $U_n$ is open. The nabla product, $\nabla X^{\omega }$, is obtained from $\square X^{\omega }$ by quotienting out mod-finite. The weight of $X$, $w\left(X\right)$, is the minimal size of a base, while $d=cof{\omega }^{\omega }$.

It is shown that there are specific compact spaces $X$ such that $\square X^{\omega }$ and $\nabla X^{\omega }$ are not $D$, but in general:

(1) $\square X^{\omega }$ and $\nabla X^{\omega }$ are hereditarily $D$ if $X$ is scattered and either hereditarily paracompact or of finite scattered height, or if $X$ is metrizable (and $w\left(X\right)\le d$ for $\square X^{\omega }$);

(2) $\nabla X^{\omega }$ is hereditarily $D$ if $X$ is first countable and $w\left(X\right)\le {\omega }_1$, or consistently if $X$ is first countable and $\left|X\right|\le c$, or $w\left(X\right)\le {\omega }_1$; and

(3) $\square X^{\omega }$ is $D$ consistently if $X$ is compact and either first countable or $w\left(X\right)\le {\omega }_1$.