  A global $U(1)_\text{PQ}$ symmetry is protected from gravitational effects in
the s-confining $SU(N)^k$ product group theory with $A+4Q +N\overline{Q}$
matter. If the $SU(4)$ family symmetry is gauged and an appropriate tree-level
superpotential is added, then the dynamically generated superpotential
spontaneously breaks $SU(4)\times U(1)_\text{PQ} \rightarrow SU(3)_c$ and
produces a QCD axion. Small values of the $CP$-violating $\theta$ parameter are
then possible without any fine-tuning, as long as the product group is suitably
large. By introducing a second copy of the s-confining $SU(N)$ product group
also coupled to the gauged $SU(4)$, we find that values as small as $N=7$ are
consistent with $\bar\theta<10^{-10}$, even under the pessimistic assumption
that the dominant contribution to the axion quality is at tree level.
