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.

So now we have the clear problem.

Why are they lying, and what is the truth which they are trying so hard to conceal?
Let us try, Watson, you and I, if we can get behind the lie and reconstruct the truth.
"How do I know that they are lying?
Because it is a clumsy fabrication which simply could not be true.