Rene Harris broke her arm after slipping on what she called a "wet place" on the grand stairs. I don't recall off hand in which account she claims that it was from the oily residue of a teacake. Maybe it's her ubiquitous Liberty magazine story. But I do know that in a 1940s radio interview she said it was "a wet place where something had been mopped up."
Of course, as happened with well known survivors who were regularly called upon to relate their experiences, details Rene gave tended to vary from one telling to the next.
As to her being embarrassed by her fall - not Rene! She had too much of a devil-may-care personality for that. Besides she was in great pain. That's what her mind was on. Actually, there was no reason to be embarrassed. Quite the opposite. Rene said her fall became the talk of the voyage, that fellow passengers sent round cards and notes of sympathy to her cabin and that when she and her husband appeared in the restaurant for dinner, people crowded around their table offering their well wishes for her recovery.
An interesting aside is that Rene, in "My Scrambled Autobiography," her unpublished memoir, claims she was accident prone. But whatever her predicament, she was all smiles - she radiated cheer her whole life. In her manuscript she said she had previously broken her arm after slipping on an icy sidewalk and that in later years she broke her leg. As a theatre producer, you can imagine the jokes she made to friends about that!