I asked a simillar question with the weaker restriction:
On the equation $a^4+b^4+c^4=2d^4$ in coprime positive integers $a\lt b\lt c$ such that $a+b\ne c$
.
I couldn't find any solutionssolution to this equation. And if $a^2+2ab+b^2=c^2$, then $c>d=a+b$.
Main question: Find some solutions to the equation $a^4+b^4+c^4=2d^4$ in natural numbers with $a<b<c<d$.
Thanks for advance.