  We prove that the $L^1$-algebra of any non-Kac type compact quantum group
does not satisfy operator biflatness. Since operator amenability implies
operator biflatness, this result shows that any co-amenable, non-Kac type
compact quantum group gives a counter example to the conjecture that $L^1(\Gb)$
is operator amenable if and only if $\Gb$ is amenable and co-amenable for any
locally compact quantum group $\Gb$. The result also implies that the
$L^1$-algebra of a locally compact quantum group is operator biprojective if
and only if $\Gb$ is compact and of Kac type.
