A Godefroy—Kalton principle for free Banach lattices

Motivated by the Lipschitz-lifting property of Banach spaces introduced Godefroy and Kalton, we consider lattice-lifting property, which is an analogous notion within category lattices lattice homomorphisms. Namely, a X satisfies if every homomorphism to having bounded linear right-inverse must have right-inverse. In terms free lattices, this can be rephrased into following question: embed they generate as lattice-complemented sublattice? We will provide necessary conditions for show that shared with 1-unconditional basis well lattices. The case C(K) also analyzed.