The equation $x^T A x = (x^2)^T A x^2$

Question

Let $x$ be a positive vector and let $x^2$ denote the vector whose elements are the squares of the elements of $x$. Let $A$ be a positive symmetric matrix. Is it true that if $$x^T A x = (x^2)^T A x^2$$ then $x$ is the all-ones vector?

Answer 1

Consider $A=I_2$ and $x=(a,b)$ with $a^2+b^2=a^4+b^4$, so for example $a=\sqrt{0.5},b=\sqrt{1+\sqrt{2}}/\sqrt{2}$.