The establishment of collision-free and live vehicle motion is a prominent problem for many traffic systems. Past work studying this problem in the context of guidepath-based and free-range vehicular systems has implicitly assumed that its resolution through maximally permissive supervision is NP-hard, and therefore, it has typically pursued suboptimal (i.e., more restrictive) solutions. The work presented in this technical note offers formal proof to this implicit assumption, closing the apparent gap in the existing literature. In the process, it also derives an alternative proof for the NP-hardness of maximally permissive liveness-enforcing supervision in Linear, Single-Unit Resource Allocation Systems, that is more concise and more lucid than the currently existing proof.