Background: The equation

$$a^4+b^4+c^4=2d^4$$

has infinitely many positive integral solutions if we take $c=a+b$ and $a^2+a\cdot b+b^2=d^2$.

Main problem: Are there any positive integral solutions to the equation

$$a^4+b^4+c^4=2d^4$$

with $a\lt b\lt c\ne a+b$.