  We study entanglement generation in a quantum network where repeater nodes
can perform $n$-qubit Greenberger-Horne-Zeilinger(GHZ) swaps, i.e., projective
measurements, to fuse $n$ imperfect-Fidelity entangled-state fragments. We show
that the distance-independent entanglement distribution rate found previously
for this protocol, assuming perfectly-entangled states at the link level, does
not survive. This is true also in two modified protocols we study: one that
incorporates $l \to 1$ link-level distillation and another that spatially
constrains the repeater nodes involved in the swaps. We obtain analytical
formulas for a GHZ swap of multiple Werner states, which might be of
independent interest. Whether the distance-independent entanglement rate might
re-emerge with a spatio-temporally-optimized scheduling of GHZ swaps and
multi-site block-distillation codes remains open.
