Diamond on ladder systems and countably metacompact topological spaces

The property of countable metacompactness of a topological space gets its importance from Dowker’s 1951 theorem that the product of a normal space X with the unit interval is again normal iff X is countably metacompact. In a recent paper, Leiderman and Szeptycki studied $\mathrm{\Delta }$-spaces, which are a subclass of the class of countably metacompact spaces. They proved that a single Cohen real introduces a ladder system $L$ over the first uncountable cardinal for which the corresponding space $X_L$ is not a $\mathrm{\Delta }$-space, and asked whether there is a ZFC example of a ladder system $L$ over some cardinal $\kappa$ for which $X_L$ is not countably metacompact, in particular, not a $\mathrm{\Delta }$-space. We prove that an affirmative answer holds for the cardinal $\kappa =cf({\mathrm{\beth }}_{\omega +1})$. Assuming ${\mathrm{\beth }}_{\omega }={\mathrm{\aleph }}_{\omega }$, we get an example at a much lower cardinal, namely $\kappa =2^{2^{2^{{\mathrm{\aleph }}_0}}}$, and our ladder system $L$ is moreover $\omega$-bounded.