I've found that the Edge Chromium prompts for the proxy authorization when try to open a non-secured web-site in Private Mode. It does not provide an option to use company's SmartBadge for proxy authorization in Private Mode as the username/password option is the only offered. It works seamlessly when opening a secured web-site (https://) with no additional authorization needed at all. It also works fine in regular mode for both secured and non-secured web-sites.
I've got the case 15259 opened but suggested to post here.
I am using Microsoft Edge: Version 80.0.361.32 (Official build) beta (64-bit).