The solution of the SDE in question is an example of a time-inhomogeneous diffusion process that is stopped/killed/terminated when it first hits the origin. For pathwise uniqueness until time $t \wedge \tau$, Theorem 3.7 in Chapter 5 of the following classic should do the trick.

*Ethier, Stewart N.; Kurtz, Thomas G.*, Markov processes. Characterization and convergence, Wiley Series in Probability and Mathematical Statistics. New York etc.: John Wiley & Sons. X, 534 p. \textsterling 49.10 (1986). ZBL0592.60049.

In this setting the theorem precisely states the following.

**Theorem** (Pathwise Uniqueness). Let $U \subset \mathbb{R}$ be an open set, let $T>0$, and suppose that there exists a constant $L$ such that $$
|b(t,x)-b(t,y)| \vee |a(t,x)-a(t,y)| \le L |x-y| \qquad 0 \le t \le T \quad x,y \in U \;.$$
Given two solutions $X_t, Y_t$ of the SDE, let $$
\tau = \inf\{ t \ge 0 \mid X_t \notin U~~ \text{or}~~ Y_t \notin U \} \;.
$$ Then $P[X_0 = Y_0] = 1$ implies that $P[X_{t \wedge \tau} = Y_{t \wedge \tau}~~\text{for} ~~0 \le t \le T] = 1$.

To be sure, choose the open set to be $U = \{ x > 0 \}$. The proof of this theorem is standard, and involves computing the mean-squared difference between $X_{t \wedge \tau}$ and $Y_{t \wedge \tau}$, applying Lipschitz continuity of the coefficients, and then invoking Grönwall's inequality.