I got a bit further. It seems to be a security issue.
I found that although the router mode is set to site domains and the SSL is available and trusted, when I click on the WP Admin or Open Site Buttons, Local does not open the page using HTTPS but using plain HTTP. This seems to work fine in Firefox but unfortunately not in Edge. And as edge hides the HTTP:// part of the URL, I did not notice it.
When I change the URL to include HTTPS, it works just fine 
So, I am guessing that this is a small bug in Local.