For the Jarlskog invariant $J_{CP}^l$ in the lepton sector (defined in the same way as the one in the quark sector, see [@jcp]) we find \[eq:jcpl\] J\_[CP]{}\^l&& - 16 ( (n) + )  . As one can see, the first contribution arises from the neutrino sector, while the second one is due to the charged lepton sector. In [@HKL10] only the second term in Eq.(\[eq:jcpl\]) is present (at this level in $\la$), because $d_\nu \neq 0$ is only induced at the relative order $\la^4$.

Assuming that the value of the CP phase $\delta^l$ is dominated by the contribution from the neutrino sector,[^6] we find $\sin\delta^l \approx -\mathrm{Im}(n)/|n|$. Using this and that $\theta_C \approx \tilde{x}_2/ y_s \la$, see section \[recap\], we get \[abound\] |a|    (r + ) |\^l| . For $r \approx 0.2$ (which implies $\theta_{13}^l \approx 0.15$) and CP conservation $|\cos \delta^l|=1$ this bound on the deviation from maximal atmospheric mixing translates into $0.38 \lesssim \sin^2 \theta_{23}^l
\lesssim 0.64$, which is comparable to the $3 \, \sigma$ range quoted in [@fogli]. Obviously, for a non-trivial CP phase this bound will be tighter. For the prospects of measuring $\delta^l$ in the mid-term future, see for example [@NOvA]. Similarly, we get for 