  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.
