When debugging your site using HTTPS with IIS Express or dotnet run, Browser Link injects scripts from the HTTP scheme. This breaks pages, because you get mixed content security warnings. Ideally Browser Link should be using HTTPS as the default. Additionally, if using Content Security Policy (CSP) with the upgrade-insecure-requests directive, then the BrowserLink URL's are switched to HTTPS which fails.
Also, it would be nice if BrowserLink could tell you which port it's using or if this could be configured, so that those ports can be added to any Content Security Policy (CSP).
Related Issue #490 already shows that Browser Link does not work with HTTPS at all.