What is the main conflict in Whose Body?: A Lord Peter Wimsey Novel by Dorothy L. Sayers?

Lord Peter Wimsey is headed for a book auction when he learns from his mother that a corpse has been found in the bathtub of a local architect. Peter, an amateur sleuth, promptly makes arrangements to send his manservant, Bunter, to the auction in his stead and to go to the scene of the corpse. He arrives while the police are gone and the owner of the house, Mr. Thipps, grants Peter access to the bathroom. There he finds a nude man wearing nothing but a pair of expensive glasses, carefully arranged in the tub. The police soon come to the conclusion that Mr. Thipps, his servant named Gladys, her boyfriend William, or some combination of the three, are responsible for the man's death.