$xy^2x^{-1}=y^3$ implies $[xyx^{-1},y]=1$

Question

I want to show that if $x,y$ are elements of an hyperbolic group $G$, and $xy^2x^{-1}=y^3$, then $[xyx^{-1},y]=1$. I tried to show that by using triangle thinness but I reached nothing. Another observation is that calling $z=xyx^{-1}$, then we have that $z^2$ lays in the centraliser of $y$, also $y^3$ lays in the centraliser of $z$, and we want to show that $z$ is in the centraliser of $y$. I tried to use the fact that centralisers of elements of infinite order must be virtually cyclic in $G$ but I’m not sure that $y$ must have infinite order.