Given a (reduced) locally compact quantum group
A
A
, we can consider the convolution algebra
L
1
(
A
)
L^1(A)
(which can be identified as the predual of the von Neumann algebra form of
A
A
). It is conjectured that
L
1
(
A
)
L^1(A)
is operator biprojective if and only if
A
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
)
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.