The paracompactness conjecture for singular cardinals

Let $\kappa$ be a singular cardinal, and let ${}^{<\kappa}\square\,2^\kappa$ denote the corresponding middle box product space. The paracompactness conjecture for singular cardinals. No singular cardinal $\kappa$ is paracompact. Moreover, not even the space

$${}^{<\kappa}\square\,2^\kappa$$

is paracompact. The preceding results rule out paracompactness for infinite successor cardinals, while the singular-cardinal case remains open; proving this conjecture would support the view that paracompactness is a large cardinal notion.