  Given a (reduced) locally compact quantum group $A$, we can consider the
convolution algebra $L^1(A)$ (which can be identified as the predual of the von
Neumann algebra form of $A$). It is conjectured that $L^1(A)$ is operator
biprojective if and only if $A$ is compact. The "only if" part always holds,
and the "if" part holds for Kac algebras. We show that if the splitting
morphism associated with $L^1(A)$ being biprojective can be chosen to be
completely positive, or just contractive, then we already have a Kac algebra.
We give another proof of the converse, indicating how modular properties of the
Haar state seem to be important.
