Launcher is still temporarily offline after logging in and out of a Mojang account while a Microsoft account is logged in

Clones MCL-16349.

The bug

Upon logging out and then logging back in of a Mojang account while a Microsoft account is logged in at the same time, the Minecraft Launcher enters offline mode for an instant after signing in (around 1 second during my tests), until reverting back to online mode.

How to reproduce

1. Open the Minecraft Launcher (connected to the Internet)

2. Make sure you are signed into a Mojang account and a Microsoft account

3. Log out of your Mojang account from the dropdown menu in the top left corner of the screen

4. Click Add account -> Mojang Login -> sign into your Mojang account
   ❌  Notice how the Minecraft Launcher appears in offline mode for an instant before switching back to online mode

Additional information

This issue was more noticeable in 2.2.1166. In 2.2.1260, a fix has been implemented. However, as said in @unknown's comment on MCL-16349, this fix is not perfect. Previously, the launcher would be offline for about thirty seconds, while now, it's typically only one second (in my tests, at least).