This result has been disproven.
Let $x_1,x_2,x_3,x_4$ be an arithmetic progression and suppose that $$x_1^3, \quad x_1^3 + x_2^3, \quad x_1^3 + x_2^3 + x_3^3, \quad x_1^3 + x_2^3 + x_3^3+x_4^4$$ are perfect squares. Prove that $x_1,x_2,x_3,x_4$ are all integers.
Since these numbers are in arithmetic progression, we must have $x_1,x_1+d,x_1+2d,x_1+3d$ are our four terms. Also, $x_1$ must either be a perfect square, or the cubic root of a perfect square and $x_2,x_3$ must be integers or cubic roots. I thought about proving the latter is impossible by showing that the distance between the cubic roots of integers is unique ($\sqrt[3]{a}-\sqrt[3]b)$, so that $x_4^3$ can't be an integer. How do we continue from here?