The mere sight of his excellent appetite was an assurance of success, for I had very clear recollections of days and nights without a thought of food, when his baffled mind had chafed before some problem while his thin, eager features became more attenuated with the asceticism of complete mental concentration.

Finally he lit his pipe, and sitting in the inglenook of the old village inn he talked slowly and at random about his case, rather as one who thinks aloud than as one who makes a considered statement.
"A lie, Watson--a great, big, thumping, obtrusive, uncompromising lie--that's what meets us on the threshold! There is our starting point.

The whole story told by Barker is a lie.

But Barker's story is corroborated by Mrs.Douglas.Therefore she is lying also.

They are both lying, and in a conspiracy.