As per this blog post it should be possible: https://blog.mozilla.org/security/2020/04/14/expanding-client-certificates-in-firefox-75/
Yet when browsing a site which asks for a client certificate it doesn't open a prompt. (On the same site the prompt happens when installing said certificate in Firefox's own store.)
This happens using Firefox 89.0.2, Windows 10. The server is nginx with ssl_verify_client optional