sizes of the VEV shifts as in [@HKL10] apart from those of the flavons $\Phi^\nu_2$ and $\Phi^\nu_{3'}$, as explained.

Relating the VEVs
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As discussed in one of the appendices of [@HKL10] it is possible to add driving fields whose $F$-terms give rise to relations between different VEVs, relate the latter to explicit mass scales as well as enforce the spontaneous breaking of the flavour symmetry. In [@HKL10] two such fields $V_0 \sim
({\bf 1}, 0)$ and $V_2 \sim ({\bf 2}, -8)$ under $(S_4,U(1))$ have been proposed. Adding the field $\eta$ does not disturb the relation induced by the $F$-term of $V_0$, however it leads to problems with the field $V_2$. From the $F$-terms of the latter one can derive two independent equations relating the VEVs $\varphi^{\nu}_1$, $\wt\varphi^u_2$, $\varphi^d_2$ with an explicit mass scale. The contribution to the $F$-term of $V_2$ coming from \[V2dang\] V\_2 \^d\_2 is larger (it is of order $\la^5$) than the leading contribution without the field $\eta$ (which is of order $\la^8$) and strongly perturbs the results achieved before. In order to avoid this, we consider instead the field $V_1$ which carries the same $U(1)$ charge as $V_2$, but is a trivial singlet under $S_4$: $V_1 