Distribution of entanglement in two-dimensional square grid network

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 the modified protocol we study - one 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.