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 $a+b\ne c$

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I couldn't find any solutions 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.