  The three-dimensional pure quantum gravity with negative cosmological
constant is supposed to be dual to the extremal conformal field theory of
central charge $c=24k$ in two dimensions. We employ the conformal bootstrap
method to analyze the extremal CFTs, and find numerical evidence for the
non-existence of the extremal CFTs for sufficiently large central charge ($k
\ge 20$). We also explore near-extremal CFTs, a small modification of extremal
ones, and find similar evidence for their non-existence for large central
charge. This indicates, under the assumption of holomorphic factorization, the
pure gravity in the weakly curved AdS$_3$ do not exist as a consistent quantum
theory.
