Additivity properties of topological diagonalizations

In a work of Just, Miller, Scheepers and Szeptycki it was asked whether certain diagonalization properties for sequences of open covers are provably closed under taking finite or countable unions. In a recent work, Scheepers proved that one of the properties in question is closed under taking countable unions. After surveying the known results, we show that none of the remaining classes is provably closed under taking finite unions, and thus settle the problem. We also show that one of these properties is consistently (but not provably) closed under taking unions of size less than the continuum, by relating a combinatorial version of this problem to the Near Coherence of Filters (NCF) axiom, which asserts that the Rudin-Keisler ordering is downward directed.