  Recently Liu and Leutheusser argued that the operators in the interior of a
black hole can be reconstructed by the half-sided evolution of the non-local
operators in the boundary CFT which are dual to the bulk operators outside the
black hole horizon. However, for an AdS$_2$ black hole in equilibrium with a
finite temperature bath, the continuous exchange of Hawking quanta, emitted by
the black hole, and the bath quanta give rise to an information paradox. The
resolution of this paradox requires transferring the ownership of the black
hole interior from the boundary CFT to the bath at late time. As a result, the
reconstruction method proposed by Liu and Leutheusser must cease to reconstruct
the black hole interior after the Page time using the boundary CFT. In this
paper, we demonstrate this by showing that the relevant half-sided translations
are trivial at late time.

High Energy Physics - Theory;General Relativity and Quantum Cosmology