A new survey estimates that about a quarter of American workers have been contacted about politics by their bosses or supervisors — and that nearly two-thirds of those contacted have gotten some form of political pressure.

For some, this pressure came from a message they disagreed with or that made them uncomfortable. For others, potential consequences to the company's business due to politics — like plant closings, layoffs, or reduced hours — were mentioned.

Overall, Alex Hertel-Fernandez, a Harvard PhD student who commissioned the survey, estimates that "somewhere between 3 percent and 10 percent of all US employees — about 4 to 14 million Americans — are experiencing intimidating forms of political contact at work."

25 percent of workers were contacted about politics by their bosses

The survey, first shared exclusively with Vox, was conducted this April by the firm SSRS. About 1,000 American workers (who weren't self-employed) were polled and asked about what they heard about politics at work from their managers.

For the majority, it was nothing at all — 75 percent of respondents said their managers kept politics out of the workplace. But that still leaves millions of Americans who have been contacted, if the survey's estimates are accurate. (In comparison, only 12 percent of workers were contacted by a union.)

The messages they got varied. Some were urged to register or vote, while others were given information on issues or candidates, as you can see in this chart:

But Hertel-Fernandez stresses that not all of these messages are bad. "Many firms have constructive political engagement programs, including nonpartisan efforts to help workers to register and turn out to vote," he writes.

14 percent of workers got at least a somewhat coercive message

The potential problem comes when employees are told what to think, or whom to vote for. As Hertel-Fernandez points out, press accounts tell of firms like Georgia Pacific (a subsidiary of Koch Industries) and Cintas either telling their employees to support certain candidates or condemning certain policies.

So he looked more closely at whether these political messages from managers were benign or coercive, by asking workers whether they agreed with the messages, whether the messages made them uncomfortable, and whether they included what he calls "economic threats" — "warnings about job loss, plant closures, or changes to wages or hours."

He found that 14 percent of workers overall got some form of coercive political message. Half of those got only a "somewhat coercive" message — one they merely disagreed with but didn't make them uncomfortable — and the other half got a "strongly coercive" message that either made them uncomfortable or included those potential adverse economic consequences:

But these messages wouldn't fly everywhere — Hertel-Fernandez points to an Oregon law, the Worker Freedom Act, as a potential model for reform. The law, pushed by organized labor in the state, blocks employers from disciplining workers if they refuse to attend non-work-related events on politics, unionization, or religion. It has, however, faced criticism from some business groups for muzzling employers' speech — but supporters emphasize that they could still hold such meetings, they just couldn't be mandatory.