= Solution
Let $X$ and $Y$ be two solutions with the same initial value and Brownian motion, and put $Z=X-Y$. <Itô formula> gives
$$
dZ_t^2=\left(2Z_t(b(X_t)-b(Y_t))+
(\sigma(X_t)-\sigma(Y_t))^2\right)dt
+2Z_t(\sigma(X_t)-\sigma(Y_t))dW_t.
$$
Stop when either process or the stochastic integral becomes large. Taking expectations, using the assumed one-sided Lipschitz bound, and then removing the localization gives
$$
\mathbb EZ_t^2\leq K\int_0^t\mathbb EZ_s^2ds.
$$
The <Gronwall inequality> yields $\mathbb EZ_t^2=0$. Thus $X_t=Y_t$ almost surely at every rational time, and path continuity makes the two processes indistinguishable. This proves pathwise uniqueness.
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