Doubt in following one step of Proof

Let G be a finite group whose order is not divisible by 3. Suppose that $(ab)^3 = a^3b^3$
for all a, b in G. Prove G must be abelian.

In the proof here, there comes a step, where we have established that $a^2b^3 = b^3a^2$.
And previously, we have also established that "Every element of G can be uniquely represented as a cube".

Thus we conclude that $a^2b^2 = b^2a^2$

This is the part I don’t understand.
If $a^2b^3 = b^3a^2$ and $a^2b = ba^2$, then how does this implies to $a^2b^2 = b^2a^2$