Does a married woman have to cover her hair while davening alone in her home?

A woman is allowed to daven at home without her hair covered. Though it would be considered 'midat ha'chasidut' to wear, as we know the story in the Talmud of the woman who never showed her hair to the walls of her house and merrited children who were talmidei chachamim Rabbi Binyamin Bemberger