Which topological properties are preserved under taking box products?

Some local network or base properties are preserved by countable box-products of topological spaces. In particular:

1) the existence of a countable $cs$-network at each point;

2) the existence of a countable $cs^*$-network at each point;

3) the existence of a countable $s^*$-network at each point;

4) the existence of an $\omega^\omega$-base at each point.

Now I recall the corresponding definitions.

Let $X$ be a topological space and $x$ be a point of $X$. A family $\mathcal F$ of subsets of $X$ is called a

$\bullet$ a $cs$-network at $x$ if for any sequence $\{x_n\}_{n\in\omega}\subset X$ convergent to $x$ and any neighborhood $O_x\subset X$ of $x$ there exists a set $F\in\mathcal F$ such that $F\subset O_x$ and $F$ contains all but finitely many elements of the sequence $(x_n)$;

$\bullet$ a $cs^*$-network at $x$ if for any sequence $\{x_n\}_{n\in\omega}\subset X$ convergent to $x$ and any neighborhood $O_x\subset X$ of $x$ there exists a set $F\in\mathcal F$ such that $F\subset O_x$ and $F$ contains infinitely many elements of the sequence $(x_n)$;

$\bullet$ an $s^*$-network at $x$ if for any sequence $\{x_n\}_{n\in\omega}\subset X$ accumulating at $x$ and any neighborhood $O_x\subset X$ of $x$ there exists a set $F\in\mathcal F$ such that $F\subset O_x$ and $F$ contains infinitely many elements of the sequence $(x_n)$;

$\bullet$ an $\omega^\omega$-base at $x$ if each set $F\in\mathcal F$ is a neighborhood of $x$ and $\mathcal F$ can be written as $\mathcal F=\{F_\alpha\}_{\alpha\in\omega^\omega}$ so that $F_\alpha\subset F_\beta$ for any elements $\beta\le\alpha$ of $\omega^\omega$ (endowed with the ccordinatewise partial order).

Those notions are studied in this paper and often appear in Topological Algebra and Functional Analysis.

For a topological space $X$ and a point $x\in X$ we have the implications:

($X$ has an $\omega^\omega$-base at $x$) $\Rightarrow$ ($X$ has a countable $s^*$-network at $x$) $\Rightarrow$

$\Rightarrow$ ($X$ has a countable $cs^*$-network at $x$) $\Leftrightarrow$ ($X$ has a countable $cs$-network at $x$).

A space is said to be hyperconnected if every two non-empty open sets have non-empty intersection.

The axiom of choice implies that the box product of hyperconnected spaces is hyperconnected.

A space $X$ is discretely generated (DG) if for every non-closed set $A \subset X$ and for every point $x \in \overline{A} \setminus A$ there is a discrete set $D \subset A$ such that $x \in \overline{D}$.

For being a convergence-type property (note that Fréchet-Urysohn and even radial spaces are DG) discrete generability is pretty weak: every scattered space is DG, every compact space of countable tightness is DG in ZFC and every monotonically normal space is DG (see Dow, A.; Tkachenko, M.G.; Tkachuk, V.V.; Wilson, R.G., Topologies generated by discrete subspaces, Glas. Mat., III. Ser. 37, No.1, 187-210 (2002). ZBL1009.54005.).

There are examples of DG spaces with a non-DG square (see Murtinová, Eva, On products of discretely generated spaces, Topology Appl. 153, No. 18, 3402-3408 (2006). ZBL1107.54006.), but in a few notable special case discrete generability is indeed preserved in box products:

THEOREM (Tkachuk, 2012): Every box product of monotonically normal spaces is discretely generated.

THEOREM (Barriga-Acosta and Hernández-Hernández, 2016): Every box product of regular first-countable spaces is discretely generated.