There was no notification for an update being ready on the client, but I noticed that one was available after I attempted to watch a live-replay and it crashed.
The readme for faf-linux says to run /update.sh perform, but that did nothing. The version that it was checking was outdated.
I managed to manually update the client using
/update-component faf-client 2024.8.0. However, I have no idea if there have been any updates to the dxvk or java component versions.
Also, when I tried to check these version numbers on the git repository, the version file had not yet been updated. It is still checking for client 2024.6.2.