"We find that in markets with high individual bargaining power (low unionization rates), pay transparency mandates lower average wages around 2%, as we find in the context of the U.S. private sector. In markets with low individual bargaining power (high unionization rates), pay transparency mandates have little to no effect on wages."

This seems to not be an argument against pay transparency so much as it's an argument for unionization. Transparent prices in any market are important for market efficiency, and that's no less true for the labor market. The mechanism for decreasing wages is "employers resist issuing individual raises because they think everyone else will want a raise too", and that's a cultural thing that will pass in a few generations. I have worked on teams with plenty of people who made more than me, and I didn't take it personally because I knew full well how brilliant and hardworking they were.