 Solution

When $\bar{s}=\bar{t}$, the prefactor is $1/2$. Therefore $P(r>r_0)\to 1/2$ as $r_0\downarrow 0$. The clocks then have identical independent continuous laws, so either is equally likely to finish first. A tie has probability zero. The limit is the interruption probability, rather than one, because non-interrupted outcomes retain the negative sentinel value.