If logged in and visiting a mapped domain, then trying to go to the logout button at the top admin bar, it will try to go to:
This will fail because the https for the mapped domain doesnt exist.
Instead it should go to:
If logged in and visiting a mapped domain, then trying to go to the customize secation at the top admin bar, loading the customize area will fail...i am guessing because it is internally trying to reach the https version of the mapped domain when instead it should be trying to visit: https://networkname.com/mappeddomainblog/wp-admin/customize.php
Please test/try and don't be shy with those points!