I asked a simillar question with the weaker restriction:

https://mathoverflow.net/questions/438196/on-the-equation-a4b4c4-2d4-in-positive-integers-a-lt-b-lt-c-such-that

.

I couldn't find any solutions to this equation. And if $a^2+2ab+b^2=c^2$, then $c>d=a+b$.

Thanks for advance.