= Solution
The pair $(B,\widetilde B)$ is a Brownian motion in $\mathbb R^{2d}$. Assume inductively that $T_{i-1}<\infty$. By the <Strong Markov property> at $T_{i-1}$, the difference
$$
D_t=B^i_{T_{i-1}+t}-\widetilde B^i_{T_{i-1}+t}
$$
is a one-dimensional Brownian motion with variance rate two, started from its current value. By <recurrence of one-dimensional Brownian motion>, it hits zero in finite time almost surely. Hence $T_i<\infty$. Induction through $i=1,\ldots,d$ proves $T_d<\infty$ almost surely.
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