Is it true that pay transparency laws reduce wages?

Here's the Economist article that makes this claim: https://www.economist.com/united-states/2023/01/05/pay-transparency-laws-do-not-work-as-advertised

Relevant parts:

>Labour advocates champion pay-transparency laws on the grounds that they will narrow pay disparities. But research suggests that this is achieved not by boosting the wages of lower-paid workers but by curbing the wages of higher-paid ones. A forthcoming paper by economists at the University of Toronto and Princeton University estimates that Canadian salary-disclosure laws implemented between 1996 and 2016 narrowed the gender pay gap of university professors by 20-30%. But there is also evidence that they lower salaries, on average. Another paper by professors at INSEAD, UNC Chapel Hill, Cornell and Columbia University found that a Danish pay-transparency law adopted in 2006 shrank the gender pay gap by 13%, but only because it curbed the wages of male employees. Studies of Britain’s gender-pay-gap law, which was implemented in 2018, have reached similar conclusions.

>[...] the effects of 13 state laws passed between 2004 and 2016 that were designed to protect the right of workers to ask about the salaries of their co-workers. The authors found that the laws were associated with a 2% drop in wages, an outcome which the authors attribute to reduced bargaining power. “Although the idea of pay transparency is to give workers the ability to renegotiate away pay discrepancies, it actually shifts the bargaining power from the workers to the employer,” says Mr Pakzad-Hurson. “So wages are more equal,” explains Ms Cullen, “but they’re also lower.”

Is this viewpoint mainstream, or the article cherry-picks studies that support their claim? Or something in between? If true, do we understand the mechanism of these results?

Thank you for explaining.

What exactly did they achieve with all this nefarious activity? The only meaningful allegation I saw in the paper is putting emphasis on physical activity. Unless they suggest that people should drink coca-cola while exercising, I don't see anything incriminating in the article.