The authors' claim is that it is cheaper than other catalytic methods that have been explored/invented to depolymerize PET into TPA monomers. These qualitative cost estimates are based on the reaction conditions (temperature, solvent, other reactants (in this case, humid air)) and the unit operations involved in the downstream separation processes that isolate the TPA product from unreacted PET. The largest hurdle that precludes widespread deployment of technologies for PET recycling, as well as those for most other plastics, occurs (way) upstream of the reaction and separation train. The highest cost is related to collecting and sorting used PET bottles and TPA-derived textiles.
Mechanical recycling or any flavor of chemical recycling (pyrolysis, hydrolysis, etc.) all suffer from the same hurdle. If the target product of the recycling process is a TPA-derived plastic (be it for clothing or soda bottles), then mechanical recycling is usually cheaper, since it produces a product that only needs to be reshaped and remolded to give shirts or jugs. Chemical recycling converts PET into its constitutive monomers, and to (re)produce a TPA-derived plastic from the monomers requires a not inexpensive (re)polymerization step, in addition to reshaping and remolding.
Chemists, even highly regarded ones like Tobin Marks, are less interested in "solving" the PET recycling issue and more interested in the fundamental chemistry involved in chemical recycling. Issues of Green Chemistry (or blurbs in phys.org) are not the appropriate reading materials to get insight into costs, scale-up, etc.. Very few, if any, academic journals are focused on such matters, and rightly so, in my opinion.