Kellner's conjecture on integer ratios of consecutive power sums

Let $S_k(n)=\sum_{j=1}^{n-1}j^k$ for positive integers $k$ and $n$. Kellner's conjecture. Let $k\ge 1$ and $n\ge 3$ be integers. Then the ratio

$$\frac{S_k(n+1)}{S_k(n)}$$

is an integer if and only if $n=3$ and $k=1$ or $3$. This conjecture concerns the classification of integer ratios of consecutive power sums; it remains open, although various partial results are known.