The book once belonged to Barry Bloomfield, Larkin's friend and bibliographer. Laurence bought a great many books owned by Bloomfield, and writes:
'Both Barry and Larkin shared an enthusiasm for old-fashioned detective fiction - Margery Allinghan, Francis Beeding, Josephine Bell, Christiana Brand, Freeman Willis Croft and all the rest. Barry had loads of it. Most of it was housed in the garage, but when I was clearing the books away after Barry's death, I asked Valerie, his widow, why there were shelves of Gladys Mitchell in the guest bedroom. Her explanation was that this was because Gladys Mitchell was Larkin's especial favourite, and she was kept in the bedroom so that she should be at hand when Larkin came to stay'.
So we know that Larkin would at least have seen the copy on display in the show. And he may even have handled it and read it in bed!
