Of course they won't. That's why there's precisely an exemption for government personnel in Chat Control.

https://europeanpirates.eu/chatcontrol-eu-ministers-want-to-...

Not just exempting themselves, but actively building and deploying end-to-end encrypted chat apps for gov only use with (non)retention and (un)transparancy fully under their own control. Tbf, we're still waiting for Von Der Leyen's SMS's, and that was not even on such an app. Some can just ignore the law without repercussions.