Amount CP violation needed

It is widely known that the CP violation provided by the Standard Model ( SM )is insufficient to explain the baryon asymmetry that we observe. However, I never came across any quantitative argumentation about this fact. By that, I mean that we know that

$$\bullet$$ The baryon asymetry is measured to be $$\eta \simeq 6 \cdot 10^{-10}$$

$$\bullet$$ The CP violation coming from the CKM matrix is "predicted" ( we still need experiments to find the values of the CKM parameters ) in the SM to be $$J\simeq 3\cdot 10^{-5}$$, where $$J$$ is the Jarkslog invariant. We could also take, as quantitative measurement of the CP violation coming from the CKM matrix, the value of the mixing parameter $$\epsilon \simeq 2\cdot 10^{-3}$$ in the $$K-\bar{K}$$ system.

My question is the following. Is there any prediction of $$\eta$$ assuming that the CP violation is only coming from the CKM matrix or, conversely, is there any estimation of what should be the value of $$J$$ or $$\epsilon$$ to get the right value of $$\eta$$. I understand that it should heavily depends other parameters due to the 2 other Sakharov conditions but I wonder if any privilegied model has been chosen to make this statement that the SM do not provide enough CP violation.