  For a hyperbolic $3$-orbifold with underlying space the $3$-sphere, we obtain
a lower bound on its volume in the case that it contains an essential
$2$-suborbifold with underlying space the $2$-sphere with four cone points. Our
techniques involve computing the guts of the orbifold split along the
$2$-suborbifold via a careful analysis of its topology. We also characterize
the orbifolds of this type that have empty guts.
