 Solution

For every $a>0$, defineThe Brownian scaling theorem makes $\hat{B}$ a standard Brownian motion, and substitution shows that $(\hat{X},\hat{Y})$ satisfies the same coupled Bessel process equations from initial values $(x/a,y/a)$. Both hitting times are divided by $a^2$, so their order is unchanged. Taking $a=x$ or comparing any two pairs with the same ratio proves that $\mathbb{P}({\tau }_x>{\tau }_y)$ depends only on $x/y$.