Skolem’s “Theorem 9” and the associated equation

Question

In Diophantine Equations (page 209), Mordell writes:

[Skolem] proved that the equation $$x^5+2y^5+4z^5-10xy^3z+10x^2yz^2=1$$ has at most six solutions in integers $x,y,z$, and of these three are known, namely $(x,y,z)=(1,0,0),(-1,1,0),(1,1,-2).$

Question #1: Does anyone know if progress has been made on solving this equation since Mordell wrote that (1969)? I’ve been looking, but haven’t found anything.

Question #2: Is there a translation of Skolem’s book/paper anywhere, in which this “Theorem 9” can be found?

Answer 1

Theorem (Bremner 1977): T. Skolem's Diophantine equation $$x^5 + 2y^5 + 4z^5 − 10xy^3z + 10x^2yz^2 = 1$$ has precisely three integer solutions.

Reference: this paper in Journal of Number Theory, Volume 9, Issue 4, November 1977, Pages 499-501.