 Event type is independent of first time in exponential competing risks

For independent exponential competing times with rates $\lambda ,\gamma$, the observed first time has rate $s=\lambda +\gamma$, while death-type probability is $\lambda /s$. The joint density factors as $\lambda e^{-st}=(\lambda /s)s{e}^{-st}$. Thus the conditional observed time given type still has rate $s$. Independence of type and time generally fails for time-varying hazard ratios.