Forbidden integer ratios of consecutive power sums

Let $S_k(m):=1^k+2^k+\cdots +(m-1{)}^k$ denote a power sum. In 2011 Bernd Kellner formulated the conjecture that for $m\ge 4$ the ratio $S_k(m+1)\mathrm{/}{S}_k(m)$ of two consecutive power sums is never an integer. We will develop some techniques that allow one to exclude many integers $\rho$ as a ratio and combine them to exclude the integers $3\le \rho \le 1501$ and, assuming a conjecture on irregular primes to be true, a set of density $1$ of ratios $\rho$. To exclude a ratio $\rho$ one has to show that the Erd\H{o}s-Moser type equation $(\rho -1)S_k(m)=m^k$ has no non-trivial solutions.