Thanks to New York Public Library patron research database access, The Wikipedia Library https://wikipedialibrary.wmflabs.org/about/ which "allows active editors to access a wide range of collections of paywalled reliable sources for free", and the Internet Archive's ebook program, I followed a citation trail, from scholarly reviews of a Fletcher biography, to the biography itself, to Johnson's autobiography.
And I confirmed: it was true! And I edited https://en.wikipedia.org/wiki/James_Weldon_Johnson#Education_and_law_careers and Fletcher's bio accordingly.