This question is motivated by recent discovery of Andrew Booker+Andrew Sutherland.
Richard Guy's problem D5 in his Unsolved Problems in Number Theory contains the original question for the sum of three cubes and a few other interesting problems, e.g. the solutions of Diophantine equation $$x^3+y^3+2z^3=k,$$ where $k$ is an integer neither a cube nor twice a cube. For $k<1000$, only two candidates $k=148, 671$ are left(see Tomita Seiji's database).
Question: Is there any known computational effort for these two Diophantine equations?