"Schäffer's conjecture" on equation $1^k+2^k+\cdots+x^k=y^n$

Question

In 1956 J. J. Schäffer proved that the equation $$1^k+2^k+\cdots+x^k=y^n$$ for fixed integers $k\ge1,n\ge2$ has only finitely many solutions in positive integers unless $(k,n)\in\{(1,2),(3,2),(3,4),(5,2)\}$ [see Theorem 2, p. 27 in the file's numbering of pages]. Furthermore, he conjectured the equation has only trivial solution $x=y=1$ except $(k,n,x,y)=(2,2,24,70)$ (and excluding those four pairs of $(k,n)$).

I was able to find partial results, that the conjecture is true for

1. $k\le11$ [see this]
2. $n=2,k\le58$ [mentioned there as well]
3. $k<170$ odd and even $n$ [see this one]

The newest paper (3.) was published in 2007 and I didn't find newer articles. So here is my question, is Schäffer's conjecture still an open problem?