Let $x,y\ge 1$ be non-integer real numbers such that $\lfloor x^n\rfloor\lfloor y^n\rfloor$ is a perfect square for all natural numbers $n$. Does it follow that $x=y$?

From this question we know the condition under which $\lfloor a\rfloor\lfloor b\rfloor = \lfloor ab\rfloor$. I thought that this implies that for large $n$ it will not be the case that $\lfloor x^n\rfloor\lfloor y^n\rfloor = \lfloor x^ny^n\rfloor$, but now I do not see why.