lost cause

John Grishamのジュニア小説Theodore Boone the Abductionが面白かったので、図書館にあるTheodore Booneシリーズを全部読むことにしました。今回借りたのはTheodore Boone the Accusedです。 Theo達は学校のホームルームで、ある殺人事件の容疑者について話をしているところです。 "Guilty!" yelled Woody from the back of the room. Several others chimed in and added their agreement.
...
Theo did not vote, but instead became irritated. "This is ridiculous! How can you vote guilty before the trial has started, before we know what the witnesses will say, before anything happens? We've talked about the presumption of innocence. In our system, a person charged with a crime is presumed to be innocent until proven guilty.
...
Theo pressed on, still pretending to be indignant at his classmates' rush to judgment. "And proof beyond a reasonable doubt, remember? What's the matter with you guys?"
"Guilty!" Woody yelled again, and got some laughs.
Theo knew it was a lost cause. He said, "Okay, okay, can I go now?"
"lost cause" ですが、文脈から大体の意味は分かります。辞書で色々と他の例文も読んでみます。
・Oxford English Dictionary: A person or thing that can no longer hope to succeed or be changed for the better.: But I'm afraid I've never been one for lost causes.
・Cambridge English Dictionary: someone or something that has no chance of succeeding: used to try to get him to do some exercise but then decided he was a lost cause.
・Collins Dictionary: a cause with no chance of success: Frannie kept trying to hush him, but it was a lost cause.
・Wiktionary: (idiomatic) A cause, attempt, or effort that is hopeless or futile.: He has already made up his mind, and it's a lost cause to try to change it.
"lost cause" の意味は分かりましたが、Theoが思う "it was a lost cause" はWoodyが主張する "Guilty!" なのか、それともTheoが筋を通して "proof beyond a reasonable doubt" と言っても相手に分からせるのは無駄と思ったのでしょうか？