I don't think there is any compartment with pressure above atmospheric, but the passage quoted from the article does make it seem that way, hence the confusion. I think the pneumatic grabber creates a sealed region of less-than-atmospheric pressure around the site of the pinhole. Effectively, this is like increasing the pressure outside most of the eggshell.

In the 2020 video, the shot that starts at 55 seconds shows the whole carousel. A pair of robot arms are using pneumatic grabbers to load eggs onto the carousel, they go under the high power laser, then pass under the pipette robot arms, and finally are removed from the carousel by a similar pair of robot arms with pneumatic grabbers (it is at this last step that I suspect the fluid passes through the shell).