Relation between Cartesian vertex span and radius capture number

Determine whether the Cartesian vertex span σ□V(G) of every connected graph G is an upper bound for its radius capture number, namely whether rc(G) ≤ σ□V(G).

Background

The paper introduces the Cartesian vertex span as the maximum safety distance that two players can maintain while traversing all vertices of a graph under the specified movement rules. The authors note that this invariant resembles the cop and robber game with radius of capture, although the objectives and traversal requirements differ. They explicitly leave unresolved whether the Cartesian vertex span always bounds the radius capture number from above.