Margaret Cole is credited with writing 27 detective novels with her husband, the prominent socialist academic GDH Cole, between 1925 and 1942. In reality, as I explain in my book The Spectrum of English Murder, she probably wrote ten of the co-credited novels mostly herself, and GDH (Douglas) probably wrote seventeen of them mostly himself. (Another detective novel is officially credited to Douglas Cole alone.) Margaret Cole also is responsible for a collection of short mystery fiction.
The Murder at Crome House (1927)
Poison in the Garden Suburb (1929)
Burglars in Bucks (1930)
Dead Man's Watch (1931)
Death of a Star (1932)
Death in the Quarry (1934)
Scandal at School (1935)
Mrs. Warrender's Profession (1938) (short fiction)
Greek Tragedy (1939)
Counterpoint Murder (1940)
Knife in the Dark (1941)
So if Margaret Cole came to the rescue when the late Annie Haynes' unfinished manuscript for The Crystal Beads Murder needed completing in 1929-30, this came at a time when she was launched on her second and third detective novels, in addition to all her activities as, like her husband, a prominent English socialist intellectual. I just can't see this, nor is there anything in the writing that to me reads like Margaret Cole's fiction.
Conclusion: Highly unlikely.