The exact asymptotic of the collision time tail distribution for independent Brownian particles with different drifts

In this note we consider the time of the collision $\tau$ for $n$ independent Brownian motions $X^1_t,...,X_t^n$ with drifts $a_1,...,a_n$, each starting from $x=(x_1,...,x_n)$, where $x_1<...<x_n$. We show the exact asymptotics of $P_x(\tau>t) = C h(x)t^{-\alpha}e^{-\gamma t}(1 + o(1))$ as $t\to\infty$ and identify $C,h(x),\alpha,\gamma$ in terms of the drifts.