The web companion currently cannot load resources, like icons, because they are served from the remote extension host using the vscode-remote-scheme. There are three potentials approaches to tackle this

1. more uri-transforming. we already know that this is challenging because uris often come as css-rules and we don't look at those strings
2. patch document.createElement and have special cases for style and img tags (and others)
3. use the mutation observer to know when a remote-resource is being requested

So, far option 1 seems tricky, option 2 seems pragmatic, and option 3 seems to be the browser way. There was also an attempt use service workers to serve such resources but only the https-scheme is supported...

cc @mjbvz who has a similar problem with webviews (and explored the service worker route)