The conditions (S1) and (S2) imply hypergroup structure

Let $(\Gamma,w_0)$ be a pointed graph satisfying the conditions $(S1)$ and $(S2)$. Say that it produces a hypergroup when its associated distance distribution yields a hypergroup. Hypergroup conjecture. The conditions $(S1)$ and $(S2)$ imply that $(\Gamma,w_0)$ produces a hypergroup. The authors report that all known examples satisfying these conditions produce hypergroups, but no general proof or counterexample is given.