Tight lower bounds for equal-speed cop numbers of hypercubes

Prove that for every s>2 and every k≥0, a robber can evade k cops in the speed-(s,s) game on the hypercube Q_{2s+2k−2}, thereby establishing a matching lower bound for the stated upper bound on c_{s,s}(Q_d).

Background

Theorem on hypercube upper bounds gives c_{s,s}(Q_d)≤(d−2s+3)/2 for d≥2s, and the paper proves that this bound is tight when s=2. For speeds s>2, the authors state that matching the upper bound would require a robber strategy at the threshold dimension d=2s+2k−2. They regard this as unresolved because the speed-2 technique does not extend and the available high-speed lower-bound method works only in a restricted parameter regime.