"Class shapes not just who negotiates for higher salaries, but who pays a price for speaking up"

This conclusion is absolutely not justified by the findings. 

In every case, the authors used educational attainment as a proxy for class. Educational attainment means that you have something objective to negotiate about: I have this degree, so you can compare my salary to others with this degree; and also, I have this degree, which is an outward indicator of my potential that other employers can see just as easily as you can, so if you're not too keen to keep me on your payroll, I have other options. 

In an attempt to correct for this shortcoming, the authors used parents' educational attainment as a proxy for class, but this adjustment suffers from exactly the same shortcoming. If your parents had the leg-up in salary negotiations that educational attainment provides, you're more likely to have heard about how and why to negotiate salaries than someone with less educated parents. 

Class is shoehorned into the conclusion only because of the prevailing political biases of American academics. If the authors really wanted to know how class affects salary negotiations, they would have compared different socioeconomic strata within a given level of educational attainment or parental educational attainment, rather than to operationalise their suspected cause using a surrogate metric which already has obvious implications for the measured outcome. 

“More research is needed to replicate the findings across many diverse cultures.” Ah, now here's the real negotiating tactic. We know that blokes in Sydney use their credentials to ask for higher wages, but do we know if folks in Ouagadougou do? Break out the chequebook.