  The vertical Sato--Tate conjectures gives expected trace distributions for
for families of curves. We develop exact expression for the distribution
associated to degree-$4$ representations of $\mathrm{USp}(4)$,
$\mathrm{SU}(2)\times\mathrm{SU}(2)$ and $\mathrm{SU}(2)$ in the neighborhood
of the extremities of the Weil bound. As a consequence we derive qualitative
distinctions between the extremal traces arising from generic genus-$2$ curves
and genus-$2$ curves with real or quaternionic multiplication. In particular we
show, in a specific sense, to what extent curves with real multiplication
dominate the contribution to extremal traces.
