Paracompact subspaces in the box product topology

In 1975 E. K. van Douwen showed that if $( X_n )_{ n \in \omega }$ is a family of Hausdorff spaces such that all finite subproducts $\prod _{ n < m } X_n$ are paracompact, then for each element $x$ of the box product $\square _{n \in \omega } X_n$ the $\sigma$-product $\sigma ( x ) = \{ y \in \square _{n \in \omega } X_n : \{ n \in \omega : x (n) \neq y (n) \} \text{ is finite} \}$ is paracompact. He asked whether this result remains true if one considers uncountable families of spaces. In this paper we prove in particular the following result:

Let $\kappa$ be an infinite cardinal number, and let $( X_{\alpha } )_{\alpha \in \kappa }$ be a family of compact Hausdorff spaces. Let $x \in \square = \square _{\alpha \in \kappa } X_\alpha$ be a fixed point. Given a family $\mathcal{R}$ of open subsets of $\square$ which covers $\sigma ( x )$, there exists an open locally finite in $\square$ refinement $\mathcal{S}$ of $\mathcal{R}$ which covers $\sigma ( x )$.

We also prove a slightly weaker version of this theorem for Hausdorff spaces with ``all finite subproducts are paracompact" property. As a corollary we get an affirmative answer to van Douwen's question.