  One of the greatest challenges for models of $b \to s$ anomalies is the
necessity to produce a large contribution to a quark times a lepton current,
$J_q \times J_\ell$, and to avoid accordingly large contributions to
flavour-changing $J_q \times J_q$ and $J_\ell \times J_\ell$ amplitudes, which
are severely constrained by data. We consider a gauged horizontal symmetry
involving the two heaviest generations of all left-handed fermions. In the
limit of degenerate masses for the horizontal bosons, and in the absence of
mixing between the two heavier generations and the lighter one, such symmetry
would make $J_q \times J_q$ and $J_\ell \times J_\ell$ amplitudes exactly
flavour-diagonal. Mixing with the first generation is however inescapable due
to the CKM matrix, and the above mechanism turns out to be challenged by
constraints such as $D^0 -\bar D^0$ mixing. Nonetheless, we show that a
simultaneous description of all data can be accomplished by simply allowing for
non-degenerate masses for the horizontal bosons. Such scenario predicts
modifications in several processes that can be tested at present and upcoming
facilities. In particular, it implies a lower and upper bound for $\mathcal B
(B \to K \mu^\pm \tau^\mp)$, an asymmetry between its two charge conjugated
modes, and well-defined correlations with LFV in $\tau$ decays.
