Nichelle Nichols makes this claim in her autobiography, too, even adding that Rodgers had a dance choreographed for “Loads of Love,” as she was a better dancer than Carroll.
As far as I can tell, this is not confirmed anywhere. But it would not be the first time she was inaccurate about her relationship with Broadway. When she died, I posted the famous story about how she wanted to quit STAR TREK in 1967 because Uhura had so little to do, but a chance meeting with a Trekkie named Martin Luther King, Jr. made her change her mind. At various times, she said she was quitting to resume her Broadway career, at other times, she said she’d been offered a role in an upcoming show..but never mentioned which show.