Actually, I doubt it will ask.
I discovered, and replicated, the "split password" situation several times now.
Think about it, Windows must allow you to log on whether you have internet connectivity or not, so there is definitely a local, encrypted copy of the password on your machine itself.
What I found is this, if you change your Microsoft Account password you can, of course, use the new one to log in under Windows 10. But, until you do, if you keep using the old one to log in to your Windows 10 user account it will continue to work indefinitely. I think I kept this setup going once for about 2 months.
The local copy of the password that is used for verification has precedence and is checked first, so if you enter (or have things set up such that it's auto-entered) it and it passes, no internet verification is sought.
However, the first time you enter your new password, and local verification fails, it appears under the hood the next step is to verify against the existing Microsoft account password. If that passes, the current Microsoft account password is used to supplant the local copy and becomes the local copy as well. After this, your old password will not work.
All one needs to do to test this out is to change one's Microsoft Account password on an account that's linked to your Windows 10 User Account, but DON'T use that new password to log in on Windows 10. You'll see that you still get in on Windows 10, and will continue to do so until or unless you DO use that new password. After that, your old one will no longer work.