From a thermodynamic standpoint the idea is that those hydrogen bonds, no matter how many, can form just as easily/easier with water in an unfolded structure as well, not only in the duplex structure, to provide stabilization. Thus it imparts specificity (G>C and A>T) to those interactions more so than stability (though they do contribute to stability, just to a smaller degree than the hydrophobic effect, unless significant bias in GC% obviously ).
The paper’s main claim had nothing to do with rewriting basic knowledge of DNA stability no matter how they state it. At least, no scientist I know typically thinks about nucleic acid stability in semi hydrophobic solutions? Rather, it was about a possible mechanism of DNA recombination enzymes from data collected in non-aqueous environments that did seem quite interesting. They say:
“Recombinases must simultaneously both soften the DNA duplex, which is mainly stabilized through base stacking, as well as increase the fidelity of sequence recognition, which can only be achieved by strengthening the hydrogen bonds involved in base pairing. We earlier provided some evidence that a model system of aqueous polyethylene glycol (PEG) can achieve these 2 seemingly conflicting objectives (25, 26) and we here demonstrate that ethylene glycol ethers, in addition to promoting base unstacking, also strengthen base pairing, thereby increasing the energy difference between matched and mismatched sequences.
Our observation of base unstacking in hydrophobic or semihydrophilic water mixtures could have great general impact, hydrophobic catalysis potentially having a role in enzyme function, in particular of recombinases and DNA polymerases. Despite numerous studies of DNA in various solvents, specific longitudinal unstacking in a hydrophobic environment has to our best knowledge never been reported before.”
So based off my quick look at the paper the journalist is a farce and it seems like transferring DNA to a carefully balanced semi hydrophobic environment puts more onus on the H bonds for stabilizing and less so on the base stacking - an environment mimicked by certain DNA enzymes so they can function efficiently