It is obviously not yet implemented, but I believe the web service will send the challenge/response flow will be something like:
Challenge: web application -> browser -> credential manager API - > KeePassDX
Response: KeePassDX -> credential manager API -> browser -> web application
KeePassDX (or other credential provider) handles the cryptography without exposing the private key.
This is a simplified model and may not be entirely correct, and I could be wrong somewhere.
I am not sure it would add much additional risk if implemented properly and should prevent exposing a password. I am not sure how the API works under the hood.