$ \sup_{t \geq 0}\big( |B_t|-t^{r/2}\big) \text{ and } \sup_{t \geq 0}\big( \frac{|B_t|}{1+t^{r/2}}\big)^\rho$ have the same law?

Question

Let $r>1$ and let $\rho$ be its Holder's conjugate and I need to show that the following two random variables have the same law

$$ \sup_{t \geq 0}\big( |B_t|-t^{r/2}\big) \text{ and } \sup_{t \geq 0}\big( \frac{|B_t|}{1+t^{r/2}}\big)^\rho$$

I asked a very similar question in a previous thread

How can I show that $\sup_{t \geq 0}(B^*_t-\mu t^{r/2})=_{law} \sup_{v \geq 0}(\lambda B^*_{v}-\mu \lambda^rv^{r/2})$

and despite having understood the solution and struggling for a while with this problem I am at a loss on how to pick the right scaling factor. Please do not give full details, just some kind of a hint on the scaling factor would be very helpful. Thank you