If the paper gets traction and the model specification is indeed not robust, the first thing you'll see in a few months is something like "paper xyz: comment" driving holes in the methodology getting even more traction. Empirical microeconomics is fairly open about methodology flaws and critiques.

Also, by the way, you see pretty much the same type of thing in a ton of fMRI neuroscience, medical and psychology studies (even the ones you'll later see on NPR or ted talks). You shouldn't ever believe any one empirical result in basically anything except maybe CERN particle physics type work.