The New York Globe, Friday, April 19, 1912, mentions "Nearer, My God, to Thee"
I recently read James Bisset's Tramps and Ladies; Bisset was 2nd Officer on Carpathia. The book is available here: TRAMPS AND LADIES : SIR JAMES BISSET : Free Download & Streaming : Internet Archive
Many accounts mention how a reporter was able to smuggle himself aboard Carpathia in New York harbor; he was confined to the bridge by Captain Rostron. Bisset recalled this conversation with the reporter, who was from the New York Globe:
"And who are you, Mister? What's your story? Is it true that the Titanic's officers shot the third-class women and children dead, so that millionaires could get into the boats? Is it true that you picked up people floating on lumps of ice? Is it true that the band played 'Nearer, My God, to Thee'? Is it true that dogs were saved and children left to drown?" (Tramps and Ladies, p. 328)
Remember, at this instant in time, the only news of the Titanic disaster on shore has come from Carpathia's transmissions of survivor lists and from observation of Titanic's CQD/SOS messages. Conversely, aboard Carpathia, nobody knows more than what they saw and discussed. They have not been exposed to the shore press' rumors.
Of the reporter's questions, two really stand out: there were gunshots, and the band played 'Nearer, My God, to Thee.' These questions were not prompted by radio messages from Carpathia. They were purely based on rumor.
I've always been a believer in the "Nearer, My God, to Thee" ending for the band. But I think there is something important in this observation from Bisset. Was he recalling the question precisely as posed to him? If so, we should remember that one of the earliest survivor's accounts, that of Harold Bride, mentions "Autumn" as the last song played. When they emerged from Carpathia into New York City, the survivors would at once face an onslaught of rumor and misinformation in the press.
About the gunshots, I really do wonder, because it emerged that guns really were fired (even if not suicidally).
"Even while the boats were being lowered, the lively melodies of Yip-I-Addy-I-Aye and Alexander's Ragtime Band encouraged the people to remain cheerful." (p. 322)
I've seen "Alexander's Ragtime Band"
It's not in the I Salonisti recordings.