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Alice is a (potentially modified) browser/client. Bob is a non-modified HTTPS web server running common security practices.
Eve is collecting data from Bob from many Alices, Eve needs this information to be accurate, and Alice is happy to supply the information as they get paid. Eve however needs to verify the information was sent by bob, and not a scam attempt by Alice to make quick money.
Alice however does not want Eve to know their cookies etc and gain control over their account.
Can Alice prove to Eve that the data in a https / tls exchange came from bob, and that they didn't fake the data. And if so, can they do so without Eve stealing their session/cookies.