Along with his confirmation of the $e$-positivity of all cycle-chord graphs $θ_{ab1}$, the third author conjectured the $e$-positivity of all theta graphs $θ_{abc}$. In this paper, we establish the $e$-positivity of all clock graphs $θ_{ab2}$ by using the composition method. The key idea is to investigate the fibers of certain partial reversal transformation on compositions with all parts at least $2$.