Can the CKM matrix supply the CP asymmetry as required by baryogenesis, in principle?

No. The baryon-to-photon ratio observed in the universe is $10^{-10}$, whereas the calculations done in [1] imply that the CP violation induced in CKM matrix can at most produces ratio $10^{-20}$. Baryon asymmetry is only generated at seven loop level (a huge suppression!), and the calculation involves the sum of over 10000 Feynman diagrams. For similar reason, the yet unconfirmed leptonic CP violation cannot supply the necessary amount of CP asymmetry for baryogenesis.

[1] Nuclear Physics B Volume 287, 1987, Pages 757-775

In the quark sector of SM, CP violation comes from a complex phase in the CKM matrix which is a product of two unitary matrices Uu and Ud which diagonalize the quark mass matrices Mu and Md, respectively. In our recent investigations (arXiv:1902.08749), a general pattern of Hermitian mass matrices was diagonalized analytically. A CP-violating CKM matrix is thus derived. In some discussions of that article, SN symmetries among quark generations can give us a CP strength orders stronger than that SM can provide.

The CKM matrix is for quarks and the PMNS matrix is for leptons. In a broken SN symmetry model the strength of CPV is unfixed. An Improved Standard Model Comes with Explicit CPV and Productive of BAU. JMP, 11, 1157-1169.

The CPV strength characterized by Jarlskog invariant is much stronger than the value given by present standard model. Thus, there could have some BAU-productive stages in the early universe in which the CPV strength was much stronger than what we see nowadays.