Looks like they're getting closer to using hCaptcha rather than ReCaptcha [1] for the Privacy Pass service, which could be a trial of hCaptcha before rolling it out on all of the verification pages.
Why do they even need a third party captcha implementation? They could roll their own.
I work at hCaptcha so I can't speak to any particular arrangements. We provide data labeling services and use the captcha as a means to lower costs for annotated datasets. To that end, we intend to open source most of our captcha stack and would be an obvious choice for any privacy-concerned company rolling their own captcha.