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 Ined'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).