I am unable to understand what happens if the watermarked output goes as input to another agent. Let's say we asked Claude a question and got a watermarked response. If we pick that response and append it to the question we're asking ChatGPT, then will it answer or refuse to do so?

If that's the case, then it's a brilliant strategy by the labs to cut down cross-AI usage and just stick to one model. But I'm pretty sure this won't be the case.