"The Murder at Silchester Post Office" is Austin Philips' fictional reconstruction of the September, 1900 murder of George Fell, caretaker at the Birkenhead Post Office. Philips had written about this crime earlier, as part of his 1907 non-fiction Strand article "Crime in the Post Office."
The real murder was never solved. In this story, Philips imagines how the crime might have taken place, and also proposes a solution to the crime.
I wouldn't be surprised if Philips' solution was true in general outline, though of course he couldn't name any specific person. Note that---as in real life---this murder is investigated by the local police, not by Post Office detectives.