We combine the large-$c$ ST modular bootstrap equations with the Cardy
formula for the asymptotic growth of the density of states to prove that any
$2d$ unitary, compact, conformal field theory (CFT) with no higher spin
conserved currents leads to conflicting inequalities whenever the entire
spectrum of non-trivial primaries lies above the BTZ threshold. As a
consequence, the holographic dual of $3d$ pure gravity, if it exists, cannot be
a $2d$ CFT. Consistent solutions of ST bootstrap equations require additional
primaries lying below the BTZ threshold. The lowest non-trivial primary
necessarily has odd spin