Higher speed = slower clocks, but higher gravity (which is equivalent to higher acceleration) also = slower clocks.
The technical specs for GPS satellites give instructions on how to account for the satellites’ clocks both slowing down (because of their speed) and speeding up (because they are under lower gravity): in that case, the effect due to gravity is greater (i.e. overall the clocks run faster), so the clocks have to be artificially slowed to stay sync’d to clocks on Earth.
I don’t believe they’re testing Einstein’s theory of relativity, though I’m not sure what they are testing. This is a really expensive and cutting edge way to prove something already proven… and maybe that is the point, but scientists are usually working at the bleeding edge and aren’t spending a lot of hard-obtained money and equipment to do publicity stunts proving things we already know. More likely, they are using Einstein’s theory to prove their amazing new clock technology works–but it’s not my field and I’m not sure if that’s the kind of thing they would do. Something like this would be very sensitive to sway, however, as buildings aren’t quite perfect static structures, especially in earthquake prone areas. I’m curious how they account for it.