  We study compactness properties of the set of conformally flat singular
metrics with constant, positive sixth order Q-curvature on a finitely punctured
sphere. Based on a recent classification of the local asymptotic behavior near
isolated singularities, we introduce a notion of necksize for these metrics in
our moduli space, which we use to characterize compactness. More precisely, we
prove that if the punctures remain separated and the necksize at each puncture
is bounded away from zero along a sequence of metrics, then a subsequence
converges with respect to the Gromov--Hausdorff metric. Our proof relies on an
upper bound estimate which is proved using moving planes and a blow-up
argument. This is combined with a lower bound estimate which is a consequence
of a removable singularity theorem. We also introduce a homological invariant
which may be of independent interest for upcoming research.
