Schäffer's conjecture on perfect powers of power sums

For integers $k\geq1$, $n\geq2$, and $x,y$, define

$$S_k(x):=1^k+2^k+\frac{\cdots}{}+x^k.$$

Let

$$S:=\{(1,2),(2,3),(3,4),(5,2)\}.$$

Schäffer's conjecture. If $(k,n)\notin S$, then the equation

$$S_k(x)=y^n$$

has only one non-trivial solution, namely $(k,n,x,y)=(2,2,24,70)$.

Schäffer proved that outside the exceptional set $S$ there are only finitely many solutions, but his proof was ineffective. The conjecture asserts the precise complete list of non-trivial solutions and remains open.