  We show that the class of the locus of hyperelliptic curves with $\ell$
marked Weierstrass points, $m$ marked conjugate pairs of points, and $n$ free
marked points is rigid and extremal in the cone of effective codimension-($\ell
+ m$) classes on $\overline{\mathcal{M}}_{2,\ell+2m+n}$. This generalizes work
of Chen and Tarasca and establishes an infinite family of rigid and extremal
classes in arbitrarily-high codimension.
