The deterministic $k$-server conjecture states that there is a $k$-competitive deterministic algorithm for the $k$-server problem for any metric space. We show that the work function algorithm is $3$-competitive for the $3$-server problem on circle metrics, a case left open by Coester and Koutsoupias (2021). Our analysis follows the existing framework but introduces a new potential function which may be viewed as a relaxation of the counterpart by Coester and Koutsoupias (2021). We further notice that the new potential function and many existing ones can be rewritten in a canonical form. Through a computer-aided verification, however, we find that no such canonical potential function can resolve the deterministic $3$-server conjecture for general metric spaces under the current analysis framework.