A Note on Operator biprojectivity of compact quantum groups

Given a (reduced) locally compact quantum group A, we can consider the convolution algebra L1(A) (which can be identified as the predual of the von Neumann algebra form of A). It is conjectured that L1(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 L1(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.