  We give a full characterization of circle homeomorphisms which admit a
homeomorphic extension to the unit disk with finite bi-Sobolev norm. As a
special case, a bi-conformal variant of the famous Beurling-Ahlfors extension
theorem is obtained. Furthermore we show that the existing extension techniques
such as applying either the harmonic or the Beurling-Ahlfors operator work
poorly in the degenerated setting. This also gives an affirmative answer to a
question of Karafyllia and Ntalampekos.
