On stronger conjectures that imply the Erdős–Moser conjecture

The Erdős–Moser conjecture states that the Diophantine equation Sk(m)=mk, where Sk(m)=k1+k2+⋯+k(m−1), has no solution for positive integers k and m with k⩾2. We show that stronger conjectures about consecutive values of the function Sk, that seem to be more naturally, imply the Erdős–Moser conjecture.