It has been conjectured by Euler that this equation has no solutions in positive integers when $n\geq 4$.

When $n=4$, this was disproved by Elkies in the paper [Elkies, On A4+B4+C4=D4] in a very strong way  : he proves that the rational points of this K3 surface are dense in the real points for the euclidean topology.

I do not know anything about the $n\geq 5$ case.

It has been conjectured by Euler that this equation has no solutions in positive integers when $n\geq 4$.

When $n=4$, this was disproved by Elkies in the paper [Elkies, On A4+B4+C4=D4] in a very strong way: he proves that the rational points of this K3 surface are dense in the real points for the euclidean topology.

When $n\geq 5$ is odd, your surface contains lines, for instance the line $Z-T=X+Y=0$. Consequently, it has infinitely many rational points. Of course, this does not disprove Euler's conjecture, that required positive integers.