I am interested in finding more details about first collision of the 3 simple symmetric random walks. Let me make the problem precise: let $ \{ S_n^{(T)} : n \geq 0 \}, \{ S_n^{(M)} : n \geq 0 \} $ and $ \{ S_n^{(B)} : n \geq 0 \} $ be 3 independent simple symmetric random walks starting at $ 2, 0 $ and $ -2 $ respectively. Let $ \tau $ be the first collision of any pair of these random walks, i.e., \begin{align*} \tau & = \inf \{ n \geq 1 : S_n^{(T)} = S_n^{(M)} \text{ or } S_n^{(T)} = S_n^{(B)} \text{ or } S_n^{(M)} = S_n^{(B)} \} \\ & = \inf \{ n \geq 1 : S_n^{(T)} = S_n^{(M)} \text{ or } S_n^{(M)} = S_n^{(B)} \} \end{align*} as the random walks cannot cross each other without first colliding. Clearly, $ \tau < \infty $ almost surely as any pair simple symmetric (with same parity) will almost surely collide. What I am interested in is \begin{equation*} \mathbb{P} ( \tau \geq n ) . \end{equation*}
Clearly $ \mathbb{P} ( \tau \geq n ) \leq C /\sqrt{n} $ from the collision of any pair of random walks. What I am able to find from the literature (Fontes, Isopi, Newmann & Ravishankar (2003)), $ \mathbb{P} ( \tau \geq n ) \leq C_1/n $. This uses FKG on the path space of random walks.
I have simulated this quite extensively going up-to $10^9$ iterations. If we plot the logarithm of tail probability with logarithm of time. This plot shows a linear with slope very close $ -3/2 $. So, I believe that the tail probability will be behaving like $ n^{-3/2} $ maybe with some lower order (logarithmic?) correction factors. This being such a classical problem, I am wondering if there is something more is known. Any reference or any suggestions are most welcome.