Rigidity (reducedness and matrix-symmetry) makes $Γₙ$ act on the gates. The index of a gate stabiliser is at most the orbit size, so by Dixon and Mortimer the stabiliser of every internal gate contains the alternating group fixing fewer than $κ$ points on each side; reducedness upgrades it to the symmetric group, giving supports with sides of size $≤ κ - 1$ by induction. The $⊆$-least such support of a gate is mapped equivariantly by $Γₙ$, so its orbit is a lower bound for the orbit of the gate; a binomial estimate then bounds its total size by $κ - 1$.