to a relation between the VEV of $\eta$ and $\varphi^d_2$, $\varphi^d_3$ and $\varphi^\nu_1$. Choosing $M_\eta \sim \la^5 M$ gives rise to $\langle \eta \rangle \approx \la^4 M$ as required for having $\theta_{13}^l$ of order $\la$.

When recomputing the shifts in the VEVs of the flavons considering a superpotential with the fields $V_0$, $V_1$ and $V_\eta$, we have to take into account three additional shifts $\wt\Delta^u_{2,2}$, $\Delta^\nu_1$ and $\Delta^\eta$ which are related to the three additional equations coming from the $F$-terms of $V_0$, $V_1$ and $V_\eta$. These do not lead to new contributions to the fermion mass matrices which cannot be absorbed into the existing ones, but are relevant for consistently solving the $F$-term equations. We note that we find an enhancement by $\la^{-1}$ of the shifts $\wt\Delta^d_{3,2}$ and $\wt\Delta^d_{3,3}$ in the VEVs of the second and third components of the field $\wt\Phi^d_3$ and of the shift $\Delta^u_{2,2}$ in the VEV of the second component of $\Phi^u_2$ \[shiftenhance\] \^d\_[3,2]{}/M = \^d\_[3,2]{} \^4 , \^d\_[3,3]{}/M = \^d\_[3,3]{} \^4 \^u\_[2,2]{}/M = \^u\_[2,2]{} \^5  , compare to Eq.(5.3) in [@HKL10]. However, the shifts $\wt\Delta^d_{3,2}$ and $\wt\Delta^d_{3,3}$ are equal at this level and can thus be absorbed into the VEV $\wt\varphi^d_3$. The relevant difference between 