Domain Mapping login problem - only for Chrome and WPEngine

If Domain Mapping plugin is enabled on the site, you cannot login to it - even super-admin.
The login form just keeps reloading and there's "403 (forbidden)" error in the browser console.

However:

- this only happens with Chrome browser
- only in "normal" tab (it works fine in incognito mode)
- it works fine in Firefox, Edge, Opera but also Chrome incognito

Further testing revealed, that it also might be related to some cookies, probably even specific to WPEngine installs, but that's not 100% sure.