ELA, Volume 11, pp. 180-191, August 2004, abstract. The External Vertices Conjecture in Case n=4 S.W. Drury The determinantal conjecture of M. Marcus and G. N. de Oliveira is known in many special cases. The case of 3-by-3 matrices was settled by N. Bebiano, J. K. Merikoski and J. da Providencia. The case n=4 remains open. In this article a technical conjecture is established implying a weakened form of the determinantal conjecture for n=4.