On the Sidon tails of $\{\lfloor x^n\rfloor \}$

We prove that the tail of the sets $${\mathbf{S}}_x:=\{\lfloor x^n\rfloor :n\in \mathbb{N}\}$$ are Sidon for almost all $x\in (1,2)$. Then we prove that for all $\epsilon >0$, there exists $x\in (1,1+\epsilon )$ and $r\in (2-\epsilon ,2)$ such that ${\mathbf{S}}_x$ and ${\mathbf{S}}_r$ do not have a Sidon tail.