4.4.0
The Restart to Update... button in the Help menu closes the instance of the Emulator but doesn't complete the update.
Checking the version of the emulator before and after the attempt to restart the update via Help > About yields the version 4.4.0.
Expected the emulator to restart and the restarted instance to have the version number 4.4.1.

Repro'd on @corinagum's machine
[bug]
After investigating for a day or so I'm still not sure what the cause of the issue is. However, I did figure out that it seems to be a permissions issue.
It looks like the update.exe that is installed requires admin permissions to run, and so if you run the Emulator as an administrator, or if you launch it from a terminal that is running as administrator, then the update will properly install and restart the app on quit.
Most helpful comment
After investigating for a day or so I'm still not sure what the cause of the issue is. However, I did figure out that it seems to be a permissions issue.
It looks like the update.exe that is installed requires admin permissions to run, and so if you run the Emulator as an administrator, or if you launch it from a terminal that is running as administrator, then the update will properly install and restart the app on quit.