Usually multi-shank arrays like the one depicted in the press release are chronic, so no chamber is necessary. Instead, they are implanted directly into the brain during surgery and wired to a port on the animal's head. There's still some risk, since one still has to remove the skull above the target site and also cut and resew the dura mater, but the skin can be sewed back together after implantation and the bone will eventually regenerate. Unlike in an acute preparation with a chamber, there is no exposed dura that needs to be cleaned regularly to prevent infection, and the electrodes are inserted just once instead of repeatedly.
If the recording contacts are sufficiently dense, the total area of skull that needs to be removed to implant 1000 channels worth of arrays may actually be less than in a conventional two-chamber (maybe even single chamber) acute primate preparation. But we'll see.